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Mathematics LibreTexts

7-2. Sum and Difference Identities

Sum and Difference Identities
In this section, you will:
  • Use sum and difference formulas for cosine.
  • Use sum and difference formulas for sine.
  • Use sum and difference formulas for tangent.
  • Use sum and difference formulas for cofunctions.
  • Use sum and difference formulas to verify identities.
<figure class="small" id="Figure_07_02_001" style="color: rgb(0, 0, 0); font-family: 'Times New Roman'; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 1; word-spacing: 0px; -webkit-text-stroke-width: 0px;"> <figcaption>Mount McKinley, in Denali National Park, Alaska, rises 20,237 feet (6,168 m) above sea level. It is the highest peak in North America. (credit: Daniel A. Leifheit, Flickr)</figcaption> Photo of Mt. McKinley.</figure>

How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities, commonly used in mathematical proofs, have had real-world applications for centuries, including their use in calculating long distances.

The trigonometric identities we will examine in this section can be traced to a Persian astronomer who lived around 950 AD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. These are special equations or postulates, true for all values input to the equations, and with innumerable applications.

In this section, we will learn techniques that will enable us to solve problems such as the ones presented above. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the termformula is used synonymously with the word identity.

Using the Sum and Difference Formulas for Cosine

Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the special angles, which we can review in the unit circle shown in [link].

<figure id="Figure_07_02_008"> <figcaption>The Unit Circle</figcaption> Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B. </figure>

We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. See [link].

Sum formula for cosine <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )=cos α cos β−sin α sin β
Difference formula for cosine <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β )=cos α cos β+sin α sin β

First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. See [link]. Point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is at an angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>from the positive x-axis with coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos α,sin α ) and point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>Q</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is at an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>β</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>from the positive x-axis with coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos β,sin β ). Note the measure of angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mi>O</mi><mi>Q</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mo>−</mo><mi>β</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

Label two more points:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>at an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β ) from the positive x-axis with coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos( α−β ),sin( α−β ) ); and point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1,0 ). Triangle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mi>O</mi><mi>Q</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is a rotation of triangle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>O</mi><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and thus the distance from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>Q</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the same as the distance from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_07_02_002">Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B.</figure>

We can find the distance from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>Q</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>using the distance formula.

 

 

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>d</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> PQ = (cos α−cos β) 2 + (sin α−sin β) 2        = cos 2 α−2 cos α cos β+ cos 2 β+ sin 2 α−2 sin α sin β+ sin 2 β

Then we apply the Pythagorean identity and simplify.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ( cos 2 α+ sin 2 α)+( cos 2 β+ sin 2 β)−2 cos α cos β−2 sin α sin β = 1+1−2 cos α cos β−2 sin α sin β = 2−2 cos α cos β−2 sin α sin β

Similarly, using the distance formula we can find the distance from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>d</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> AB = (cos(α−β)−1) 2 + (sin(α−β)−0) 2       = cos 2 (α−β)−2 cos(α−β)+1+ sin 2 (α−β)

Applying the Pythagorean identity and simplifying we get:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ( cos 2 (α−β)+ sin 2 (α−β))−2 cos(α−β)+1 = 1−2 cos(α−β)+1 = 2−2 cos(α−β)

Because the two distances are the same, we set them equal to each other and simplify.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msqrt><mrow><mn>2</mn><mo>−</mo><mn>2</mn><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>−</mo><mn>2</mn><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>α</mi><mi>sin</mi><mtext> </mtext><mi>β</mi></mrow></msqrt></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> = 2−2 cos(α−β)   2−2 cos α cos β−2 sin αsin β=2−2 cos(α−β)        

Finally we subtract<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>from both sides and divide both sides by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>−2.</mn></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>α</mi><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>+</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mi>sin</mi><mtext> </mtext><mi>β</mi><mo>=</mo><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β )  

Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.

Sum and Difference Formulas for Cosine

These formulas can be used to calculate the cosine of sums and differences of angles.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )=cos αcos β−sin αsin β
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β )=cos αcos β+sin αsin β

Given two angles, find the cosine of the difference between the angles.

  1. Write the difference formula for cosine.
  2. Substitute the values of the given angles into the formula.
  3. Simplify.
Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles

Using the formula for the cosine of the difference of two angles, find the exact value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5π 4 − π 6 ).

Use the formula for the cosine of the difference of two angles. We have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>   </mtext><mi>cos</mi><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>α</mi><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>+</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mi>sin</mi><mtext> </mtext><mi>β</mi></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> cos( 5π 4 − π 6 )=cos( 5π 4 )cos( π 6 )+sin( 5π 4 )sin( π 6 )                    =( − 2 2 )( 3 2 )−( 22 )( 1 2 )                    =− 6 4 − 2 4                    = − 6 − 2 4

Find the exact value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 3 − π 4 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> + 6 4

Finding the Exact Value Using the Formula for the Sum of Two Angles for Cosine

Find the exact value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 75 ∘ ).

As<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> 75 ∘ = 45 ∘ + 30 ∘ , we can evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 75 ∘ ) as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 45 ∘ + 30 ∘ ). Thus,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>cos</mi><mo stretchy="false">(</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 45 ∘ + 30 ∘ )=cos( 45 ∘ )cos( 30 ∘ )−sin( 45 ∘ )sin( 30 ∘ )                        = 2 2 ( 3 2 )− 2 2 ( 1 2 )                        = 6 4 −2 4                        = 6 − 2 4

Find the exact value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 105 ∘ ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> − 6 4

Using the Sum and Difference Formulas for Sine

The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.

Sum and Difference Formulas for Sine

These formulas can be used to calculate the sines of sums and differences of angles.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )=sin α cos β+cos α sin β
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β )=sin α cos β−cos α sin β

Given two angles, find the sine of the difference between the angles.

  1. Write the difference formula for sine.
  2. Substitute the given angles into the formula.
  3. Simplify.
Using Sum and Difference Identities to Evaluate the Difference of Angles

Use the sum and difference identities to evaluate the difference of the angles and show that part a equals part b.

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 45 ∘ − 30 ∘ )
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 135 ∘ − 120 ∘ )
  1. Let’s begin by writing the formula and substitute the given angles.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>       </mtext><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>−</mo><mi>cos</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>β</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sin( 45 ∘ − 30 ∘ )=sin( 45 ∘ )cos( 30 ∘ )−cos( 45 ∘ )sin( 30 ∘ )

    Next, we need to find the values of the trigonometric expressions.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 45 ∘ )= 2 2 , cos( 30 ∘ )= 3 2 , cos( 45 ∘ )= 2 2 , sin( 30 ∘ )= 1 2

    Now we can substitute these values into the equation and simplify.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mo stretchy="false">(</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 45 ∘ − 30 ∘ )= 2 2 ( 3 2 )− 2 2 ( 1 2 )                        = 6 − 2 4
  2. Again, we write the formula and substitute the given angles.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>          </mtext><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>−</mo><mi>cos</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>β</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sin( 135 ∘ − 120 ∘ )=sin( 135 ∘ )cos( 120 ∘ )−cos( 135 ∘ )sin( 120 ∘ )

    Next, we find the values of the trigonometric expressions.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 135 ∘ )= 2 2 ,cos( 120 ∘ )=− 1 2 ,cos( 135 ∘ )= 2 2 ,sin( 120 ∘ )= 3 2

    Now we can substitute these values into the equation and simplify.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mo stretchy="false">(</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 135 ∘ − 120 ∘ )= 2 2 ( − 1 2 )−( − 2 2 )( 3 2 )                            = − 2 + 6 4                            = 6 − 2 4 sin( 135 ∘ −120 ∘ )= 2 2 ( − 1 2 )−( − 2 2 )( 3 2 )                            = − 2 + 6 4                            = 6 − 2 4
Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function

Find the exact value of<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos −1   1 2 + sin −1   3 5 ).

The pattern displayed in this problem is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β ). Let<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> cos −1 1 2  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>β</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> sin −1 3 5 . Then we can write

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>cos</mi><mtext> </mtext><mi>α</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 2 ,0≤α≤π sin β= 3 5 ,− π 2 ≤β≤ π 2

We will use the Pythagorean identities to find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mtext> </mtext><mi>α</mi><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mtd></mtr></mtable></annotation-xml></semantics></math> 1− cos 2 α        = 1− 1 4        = 3 4        = 3 2 cos β= 1− sin 2 β        = 1− 9 25        = 16 25        = 4 5

Using the sum formula for sine,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> cos −1   1 2 + sin −1   3 5 )=sin( α+β )                                        =sin α cos β+cos α sin β                                        = 3 2 ⋅ 45 + 1 2 ⋅ 3 5                                        = 4 3 +3 10

Using the Sum and Difference Formulas for Tangent

Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.

Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> sin x cos x ,cos x≠0.

Let’s derive the sum formula for tangent.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> α+β )= sin( α+β ) cos(α+β)                  = sin α cos β+cos α sin β cos α cos β−sin α sin β                  = sin α cos β+cos α sin βcos α cos β cos α cos β−sin α sin β cos α cos β   Divide the numerator and denominator by cos α cos β                  = sin α  cos βcos α  cos β + cos α  sin β cos α  cos β cos α  cos β cos α   cos β − sin α sin β cos α cos β                  = sin α cos α + sin β cos β1− sin α sin β cos α cos β                  = tan α+tan β 1−tan α tan β

We can derive the difference formula for tangent in a similar way.

Sum and Difference Formulas for Tangent

The sum and difference formulas for tangent are:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )= tan α+tan β 1−tan α tan β
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β )= tan α−tan β 1+tan α tan β

Given two angles, find the tangent of the sum of the angles.

  1. Write the sum formula for tangent.
  2. Substitute the given angles into the formula.
  3. Simplify.
Finding the Exact Value of an Expression Involving Tangent

Find the exact value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 6 + π 4 ).

Let’s first write the sum formula for tangent and substitute the given angles into the formula.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> α+β )= tan α+tan β 1−tan α tan β tan( π 6 + π 4 )= tan( π 6 )+tan( π 4 ) 1−( tan( π 6 ) )( tan( π 4 ) )

Next, we determine the individual tangents within the formula:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 6 )= 1 3 ,tan( π 4 )=1

So we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> π 6 + π 4 )= 1 3 +1 1−( 1 3 )(1)                   = 1+ 3 3 3 −1 3                   = 1+ 3 3 ( 3 3 −1 )                   = 3 +1 3 −1

Find the exact value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2π 3 + π 4 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><msqrt/></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 1+ 3

Finding Multiple Sums and Differences of Angles

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>α</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 5 ,0<α< π 2 ,cos β=− 5 13 ,π<β< 3π 2 ,find

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )
  3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )
  4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β )

We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the individual angles. To do so, we construct what is called a reference triangle to help find each component of the sum and difference formulas.

  1. To find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β ),we begin with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>α</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 5  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mfrac/></mrow></annotation-xml></semantics></math> π 2 . The side opposite<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>has length 3, the hypotenuse has length 5, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is in the first quadrant. See [link]. Using the Pythagorean Theorem, we can find the length of side<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>:</mo></mrow></annotation-xml></semantics></math>
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>a</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 + 3 2 = 5 2 ​         a 2 =16           a=4
    <figure class="small" id="Figure_07_02_003">Diagram of a triangle in the x,y plane. The vertices are at the origin, (4,0), and (4,3). The angle at the origin is alpha degrees, The angle formed by the x-axis and the side from (4,3) to (4,0) is a right angle. The side opposite the right angle has length 5.</figure>

    Since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 13  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>π</mi><mo><</mo><mi>β</mi><mo><</mo><mfrac/></mrow></annotation-xml></semantics></math> 3π 2 , the side adjacent to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>β</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>−5</mn><mo>,</mo></mrow></annotation-xml></semantics></math> the hypotenuse is 13, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>β</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is in the third quadrant. See[link]. Again, using the Pythagorean Theorem, we have

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> −5 ) 2 + a 2 = 13 2          25+ a 2 =169                        a 2 =144                          a=±12

    Since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>β</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is in the third quadrant,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>=</mo><mn>–12.</mn></mrow></annotation-xml></semantics></math>

    <figure class="small" id="Figure_07_02_004">Diagram of a triangle in the x,y plane. The vertices are at the origin, (-5,0), and (-5, -12). The angle at the origin is Beta degrees. The angle formed by the x axis and the side from (-5, -12) to (-5,0) is a right angle. The side opposite the right angle has length 13.</figure>

    The next step is finding the cosine of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and the sine of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>β</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>The cosine of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the adjacent side over the hypotenuse. We can find it from the triangle in [link]:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>α</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 4 5 . We can also find the sine of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>β</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>from the triangle in [link], as opposite side over the hypotenuse:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>β</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 12 13 . Now we are ready to evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β ).

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>+</mo><mi>cos</mi><mtext> </mtext><mi>α</mi><mi>sin</mi><mtext> </mtext><mi>β</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>                 =( 3 5 )( − 5 13 )+( 4 5 )( − 12 13 )                 =− 15 65 − 48 65                 =−63 65
  2. We can find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β ) in a similar manner. We substitute the values according to the formula.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>−</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>β</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>                  =( 4 5 )( − 5 13 )−( 3 5 )( − 12 13 )                  =− 20 65 + 36 65                 = 16 65
  3. For<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β ),if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>sin</mi><mtext> </mtext><mi>α</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 5  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>α</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 4 5 , then
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>α</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 5 4 5 = 3 4

    If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>β</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 12 13  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 13 , then

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>β</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> −12 13 −5 13 = 12 5

    Then,

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> tan α+tan β 1−tan α tan β                 = 3 4 + 12 5 1− 3 4 ( 12 5 )                 =    63 20 − 16 20 ​                =− 6316
  4. To find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β ), we have the values we need. We can substitute them in and evaluate.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> α−β )= tan α−tan β 1+tan α tan β                 = 3 4 − 12 5 1+ 3 4 ( 12 5 )                 = − 33 20 56 20                 =− 3356
Analysis

A common mistake when addressing problems such as this one is that we may be tempted to think that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>β</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are angles in the same triangle, which of course, they are not. Also note that

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )= sin( α+β ) cos( α+β )

Using Sum and Difference Formulas for Cofunctions

Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall from Right Triangle Trigonometry that, if the sum of two positive angles is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 2 ,those two angles are complements, and the sum of the two acute angles in a right triangle is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 2 ,so they are also complements. In [link], notice that if one of the acute angles is labeled as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>,</mo></mrow></annotation-xml></semantics></math> then the other acute angle must be labeled<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −θ ).

Notice also that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −θ ): opposite over hypotenuse. Thus, when two angles are complimentary, we can say that the sine of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>equals the cofunction of the complement of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.

<figure class="small" id="Figure_07_02_007">Image of a right triangle. The remaining angles are labeled theta and pi/2 - theta.</figure>

From these relationships, the cofunction identities are formed.

Cofunction Identities

The cofunction identities are summarized in [link].

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −θ ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −θ )
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −θ ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −θ )
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>csc</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −θ ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>sec</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −θ )

Notice that the formulas in the table may also justified algebraically using the sum and difference formulas. For example, using

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β )=cos αcos β+sin αsin β,

we can write

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> π 2 −θ )=cos  π 2  cos θ+sin  π 2  sin θ                  =(0)cos θ+(1)sin θ                  =sin θ
Finding a Cofunction with the Same Value as the Given Expression

Write<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 9  in terms of its cofunction.

The cofunction of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −θ ). Thus,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> π 9 )=cot( π 2 − π 9 )           =cot( 9π 18 − 2π 18 )           =cot( 7π 18 )

Write<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 7  in terms of its cofunction.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5π 14 )

Using the Sum and Difference Formulas to Verify Identities

Verifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems. Reviewing the general rules from Solving Trigonometric Equations with Identities may help simplify the process of verifying an identity.

Given an identity, verify using sum and difference formulas.

  1. Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until it matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only one side is the most efficient.
  2. Look for opportunities to use the sum and difference formulas.
  3. Rewrite sums or differences of quotients as single quotients.
  4. If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.
Verifying an Identity Involving Sine

Verify the identity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

We see that the left side of the equation includes the sines of the sum and the difference of angles.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>+</mo><mi>cos</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>β</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sin(α−β)=sin α cos β−cos α sin β

We can rewrite each using the sum and difference formulas.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>+</mo><mi>cos</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>β</mi><mo>+</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>−</mo><mi>cos</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>β</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>                                      =2 sin α cos β

We see that the identity is verified.

Verifying an Identity Involving Tangent

Verify the following identity.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></annotation-xml></semantics></math> cos α cos β =tan α−tan β

We can begin by rewriting the numerator on the left side of the equation.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> α−β ) cos α cos β = sin α cos β−cos αsin β cos αcos β                   = sin α  cos β cos α  cos β − cos α  sin β cos α  cos βRewrite using a common denominator.                   = sin α cos α − sin β cos β Cancel.                   =tan α−tan βRewrite in terms of tangent.

We see that the identity is verified. In many cases, verifying tangent identities can successfully be accomplished by writing the tangent in terms of sine and cosine.

Verify the identity:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π−θ )=−tan θ.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mo stretchy="false">(</mo><mi>π</mi><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> tan(π)−tan θ 1+tan(π)tanθ                 = 0−tan θ 1+0⋅tan θ                 =−tan θ
Using Sum and Difference Formulas to Solve an Application Problem

Let<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> L 1  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> L 2  denote two non-vertical intersecting lines, and let<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>denote the acute angle between<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> L 1  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> L 2 . See[link]. Show that

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> m 2 − m 1 1+ m 1 m 2

where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> m 1  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> m 2  are the slopes of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> L 1  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> L 2  respectively. (Hint: Use the fact that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> θ 1 = m 1  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> θ 2 = m 2 .)

<figure class="small" id="Figure_07_02_005">Diagram of two non-vertical intersecting lines L1 and L2 also intersecting the x-axis. The acute angle formed by the intersection of L1 and L2 is theta. The acute angle formed by L2 and the x-axis is theta 1, and the acute angle formed by the x-axis and L1 is theta 2. </figure>

Using the difference formula for tangent, this problem does not seem as daunting as it might.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>tan</mi><mo stretchy="false">(</mo><msub/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> θ 2 − θ 1 )        = tan  θ 2 −tan  θ 1 1+tan  θ 1 tan  θ 2        = m 2 − m 1 1+ m 1 m 2
Investigating a Guy-wire Problem

For a climbing wall, a guy-wire<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>R</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is attached 47 feet high on a vertical pole. Added support is provided by another guy-wire<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>S</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>attached 40 feet above ground on the same pole. If the wires are attached to the ground 50 feet from the pole, find the angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>between the wires. See [link].

<figure class="small" id="Figure_07_02_006">Two right triangles. Both share the same base, 50 feet. The first has a height of 40 ft and hypotenuse S. The second has height 47 ft and hypotenuse R. The height sides of the triangles are overlapping. There is a B degree angle between R and the base, and an a degree angle between the two hypotenuses within the B degree angle.  </figure>

Let’s first summarize the information we can gather from the diagram. As only the sides adjacent to the right angle are known, we can use the tangent function. Notice that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>β</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 47 50 , and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> β−α )= 40 50 = 4 5 . We can then use difference formula for tangent.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> β−α )= tan β−tan α 1+tan βtan α

Now, substituting the values we know into the formula, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>                    </mtext><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 4 5 = 47 50 −tan α 1+ 47 50 tan α 4( 1+ 47 50 tan α )=5( 47 50 −tan α )

Use the distributive property, and then simplify the functions.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mn>4</mn><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>4</mn><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 47 50 )tan α=5( 47 50 )−5 tan α                 4+3.76 tan α=4.7−5 tan α   5 tan α+3.76 tan α=0.7                           8.76 tan α=0.7                                      tan α≈0.07991               tan −1 (0.07991)≈.079741

Now we can calculate the angle in degrees.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>α</mi><mo>≈</mo><mn>0.079741</mn><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 180 π )≈ 4.57 ∘
Analysis

 

Occasionally, when an application appears that includes a right triangle, we may think that solving is a matter of applying the Pythagorean Theorem. That may be partially true, but it depends on what the problem is asking and what information is given.

Access these online resources for additional instruction and practice with sum and difference identities.

Key Equations

 
Sum Formula for Cosine <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )=cos α cos β−sin αsin β
Difference Formula for Cosine <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β )=cos α cos β+sin α sin β
Sum Formula for Sine <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )=sin α cos β+cos α sin β
Difference Formula for Sine <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β )=sin α cos β−cos α sin β
Sum Formula for Tangent <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )= tan α+tan β 1−tan α tan β
Difference Formula for Tangent <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α−β )= tan α−tan β 1+tan α tan β
Cofunction identities <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>cos</mi><mrow><mo>(</mo></mrow></mtd></mtr></mtable></annotation-xml></semantics></math> π 2 −θ ) cos θ=sin( π 2 −θ ) tan θ=cot( π 2 −θ ) cot θ=tan( π 2 −θ ) sec θ=csc( π 2 −θ )csc θ=sec( π 2 −θ )

Key Concepts

  • The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles.
  • The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle. See[link] and [link].
  • The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the second angle. See [link].
  • The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions. See [link].
  • The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that the tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the product of the tangents of the angles. See [link].
  • The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles. See [link].
  • The cofunction identities apply to complementary angles and pairs of reciprocal functions. See [link].
  • Sum and difference formulas are useful in verifying identities. See [link] and [link].
  • Application problems are often easier to solve by using sum and difference formulas. See [link] and [link].

Section Exercises

Verbal

Explain the basis for the cofunction identities and when they apply.

The cofunction identities apply to complementary angles. Viewing the two acute angles of a right triangle, if one of those angles measures<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>,</mo></mrow></annotation-xml></semantics></math> the second angle measures<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 2 −x. Then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mi>x</mi><mo>=</mo><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −x ). The same holds for the other cofunction identities. The key is that the angles are complementary.

Is there only one way to evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5π 4 )? Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.

Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>(Hint:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mo>−</mo><mi>x</mi><mo>=</mo><mo>−</mo><mi>x</mi></mrow></annotation-xml></semantics></math>)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −x )=−sinx, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> is odd.<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −x )=cos( 0−x )=cosx, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is even.

Algebraic

For the following exercises, find the exact value.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7π 12 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 12 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> + 6 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5π 12 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 11π 12 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>6</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> − 2 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> − π 12 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 19π 12 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>2</mn><mo>−</mo><msqrt/></mrow></annotation-xml></semantics></math> 3

For the following exercises, rewrite in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x+ 11π 6 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x− 3π 4 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2 sinx− 2 2 cosx

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x− 5π 6 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x+ 2π 3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 cosx− 3 2 sinx

For the following exercises, simplify the given expression.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −t )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −θ )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mi>θ</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2x )cos( 5x )−sin( 5x )cos( 2x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 2 x )−tan( 7 5 x ) 1+tan( 3 2 x )tan( 7 5 x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x 10 )

For the following exercises, find the requested information.

Given that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>a</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 3  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>b</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 4 , with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>b</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>both in the interval<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> π 2 ,π ), find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

Given that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>a</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 4 5 ,and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>b</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 3 , with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>b</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>both in the interval<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0, π 2 ), find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4 5 )( 1 3 )−( 3 5 )( 2 2 3 )= 4−6 2 15

 
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3 5 )( 1 3 )−( 4 5 )( 2 2 3 )= 3−8 2 15

 

For the following exercises, find the exact value of each expression.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos −1 (0)− cos −1 ( 1 2 ) )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos −1 ( 2 2 )+ sin −1 ( 3 2 ) )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> − 6 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> sin −1 ( 1 2 )− cos −1 ( 1 2 ) )

Graphical

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 −x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mi>x</mi></mrow></annotation-xml></semantics></math>

Graph of y=sin(x) from -2pi to 2pi.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>π</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 3 +x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 6 −x )

Graph of y=cot(pi/6 - x) from -2pi to pi - in comparison to the usual y=cot(x) graph, this one is reflected across the x-axis and shifted by pi/6.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 3 +x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 4 −x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 4 +x )

Graph of y=cot(pi/4 + x) - in comparison to the usual y=cot(x) graph, this one is shifted by pi/4.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7π 6 +x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 4 +x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>sin</mi><mi>x</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 + cosx 2

Graph of y = sin(x) / rad2 + cos(x) / rad2 - it looks like the sin curve shifted by pi/4.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5π 4 +x )

For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>x</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math> )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=sin( 4x )−sin( 3x )cos x,g( x )=sin x cos( 3x )

They are the same.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=cos( 4x )+sin x sin( 3x ),g( x )=−cos x cos( 3x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=sin( 3x )cos( 6x ),g( x )=−sin( 3x )cos( 6x )

They are the different, try<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>g</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=sin( 9x )−cos( 3x )sin( 6x ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>4</mn><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>5</mn><mi>x</mi><mo stretchy="false">)</mo><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>−</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>5</mn><mi>x</mi><mo stretchy="false">)</mo><mi>sin</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

They are the same.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> θ )=cos( 2θ ),g( θ )= cos 2 θ− sin 2 θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>tan</mi><mo stretchy="false">(</mo><mn>2</mn><mi>θ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> tan θ 1+ tan 2 θ

They are the different, try<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>g</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> θ )= 2 tanθ 1− tan 2 θ .

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo stretchy="false">)</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> sin 2 (2x) cos 2 x− cos 2 (2x) sin 2 x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>tan</mi><mo stretchy="false">(</mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> tan x−tan(2x) 1−tan x tan(2x)

They are different, try<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>g</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )= tanx−tan( 2x ) 1+tanxtan( 2x ) .

Technology

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 75 ∘ )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 195 ∘ )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 −1 2 2 , or −0.2588

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 165 ∘ )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 345 ∘ )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><msqrt/></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 2 2 , or 0.9659

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> − 15 ∘ )

Extensions

For the following exercises, prove the identities provided.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mfrac/></mrow></annotation-xml></semantics></math> π 4 )= tan x+1 1−tan x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable><mtr><mtd><mi>tan</mi><mrow><mo>(</mo></mrow></mtd></mtr></mtable></annotation-xml></semantics></math> x+ π 4 )= tanx+tan( π 4 ) 1−tanxtan( π 4 ) = tanx+1 1−tanx(1) = tanx+1 1−tanx

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>tan</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></annotation-xml></semantics></math> tan(a−b) = sin a cos a+sin b cos b sin a cos a−sin b cos b

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></annotation-xml></semantics></math> cos a cos b =1−tan a tan b

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable><mtr><mtd><mfrac><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></mfrac></mtd></mtr></mtable></annotation-xml></semantics></math> a+b ) cosacosb = cosacosb cosacosb − sinasinb cosacosb =1−tanatanb

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x+y )cos( x−y )= cos 2 x− sin 2 y

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>cos</mi><mtext> </mtext><mi>x</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> h =cos x cos h−1 h −sin x sin h h

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable><mtr><mtd><mfrac><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></mfrac></mtd></mtr></mtable></annotation-xml></semantics></math> x+h )−cosx h = cosxcosh−sinxsinh−cosx h = cosx(cosh−1)−sinxsinh h =cosx cosh−1 h −sinx sinh h

For the following exercises, prove or disprove the statements.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mo stretchy="false">(</mo><mi>u</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> tan u+tan v 1−tan u tan v

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mo stretchy="false">(</mo><mi>u</mi><mo>−</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> tan u−tan v 1+tan u tan v

True

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></mfrac></mrow></annotation-xml></semantics></math> x+y ) 1+tan x tan x = tan x+tan y 1− tan 2 x  tan 2 y

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>α</mi><mo>, </mo><mi>β</mi><mo>,</mo></mrow></annotation-xml></semantics></math> and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>γ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are angles in the same triangle, then prove or disprove<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )=sin γ.

True. Note that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> α+β )=sin( π−γ )  and expand the right hand side.

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo></mrow></annotation-xml></semantics></math> and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are angles in the same triangle, then prove or disprove<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>α</mi><mo>+</mo><mi>tan</mi><mtext> </mtext><mi>β</mi><mo>+</mo><mi>tan</mi><mtext> </mtext><mi>γ</mi><mo>=</mo><mi>tan</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>β</mi><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>γ</mi></mrow></annotation-xml></semantics></math>