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Mathematics LibreTexts

7.2 Addition and Subtraction Identities

In this section, we begin expanding our repertoire of trigonometric identities. 

Identities

The sum and difference identities

\[ \cos (\alpha - \beta) = \cos (\alpha) \cos (\beta) + \sin (\alpha) \sin (\beta)\]

\[ \cos (\alpha + \beta) = \cos (\alpha) \cos (\beta) - \sin (\alpha) \sin (\beta)\]

\[ \sin (\alpha - \beta) = \sin (\alpha) \cos (\beta) + \cos (\alpha) \sin (\beta)\]

\[ \sin (\alpha + \beta) = \sin (\alpha) \cos (\beta) - \cos (\alpha) \sin (\beta)\]

We will prove the difference of angles identity for cosine. The rest of the identities can be derived from this one.

Proof: Difference of Angles identity for Cosine

Consider two points on a unit circle:

  • \(P\) at an angle of \(\alpha\) from the positive \(x\) axis with coordinates \(\cos(\alpha),\sin(\alpha)\)
  • \(Q\) at an angle of \(\beta\) with coordinates \(\cos(\beta),\sin(\beta)\)

Notice the measure of angle \(POQ\) is \(\alpha-\beta\). Label two more points:

  • \(C\) at an angle of \(\alpha-\beta\), with coordinates \(\cos(\alpha-\beta),\sin(\alpha-\beta)\),
  • \(D\) at the point (1, 0).

Notice that the distance from \(C\) to \(D\) is the same as the distance from \(P\) to \(Q\) because triangle \(COD\) is a rotation of triangle \(POQ\).

Using the distance formula to find the distance from \(P\) to \(Q\) yields

\[ \sqrt{ \left(\cos(\alpha)-\cos(\beta) \right)^2 + \left(\sin(\alpha)-\sin(\beta) \right)^2 } \]

Expanding this

\[ \sqrt{ \cos^2(\alpha)-2\cos(\alpha)\cos(\beta) + \cos^2(\beta) + \sin^2(\alpha)-2\sin(\alpha)\sin(\beta) + \sin^2(\beta)  } \]

Applying the Pythagorean Identity and simplifying

\[\sqrt{2-2\cos(\alpha)\cos(\beta)-2\sin(\alpha)\sin(\beta)}\]

Similarly, using the distance formula to find the distance from \(C\) to \(D\)

\[ \sqrt{ (\cos(\alpha-\beta-1)^2+ (\sin(\alpha-\beta)-0)^2}\]

Expanding this

\[ \sqrt{-\cos^2(\alpha-\beta)-2\cos(\alpha-\beta)+1+\sin^2(\alpha-\beta)}\]

Applying the Pythagorean Identity and simplifying

\[ \sqrt{-2\cos(\alpha-\beta)+2}\]

Since the two distances are the same we set these two formulas equal to each other and simplify

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image305.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image307.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image309.gif

Establishing the identity

Try it Now: 1

By writing \(\cos(\alpha+\beta)\) as \(\cos(\alpha-(-\beta)\), show the sum of angles identity for cosine follows from the difference of angles identity proven above

The sum and difference of angles identities are often used to rewrite expressions in other forms, or to rewrite an angle in terms of simpler angles.

Example 1

Find the exact value of \(cos(75°\).

SOLUTION

Since File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image317.gif, we can evaluate File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image315.gif as

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image320.gif                            Apply the cosine sum of angles identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image322.gif        Evaluate

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image324.gif                                      Simply

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image326.gif

Try it Now: 2

Find the exact value of \(\sin\left(\dfrac{\pi}{12}\right)\).

Example 2

Rewrite \(\sin\left(x-\dfrac{\pi}{4} right)\) in terms of \(\sin(x)\) and \(\cos(x)\).

SOLUTION

\[\sin \left(x-\dfrac{\pi}{4} right)\]

Use the difference of angles identity for sine

\[= \sin(x)\cos \left(\dfrac{\pi}{4} \right)-\cos(x) \sin \left(\dfrac{\pi}{4} \right)\]

Evaluate the cosine and sine and rearrange

\[=\dfrac{\sqrt{2}}{2} \sin(x) - \dfrac{\sqrt{2}}{2}\cos(x)\]

Additionally, these identities can be used to simplify expressions or prove new identities

Example 3

Prove File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image337.gif.

SOLUTION

As with any identity, we need to first decide which side to begin with. Since the left side involves sum and difference of angles, we might start there

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image339.gif                                         Apply the sum and difference of angle identities

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image341.gif        

Since it is not immediately obvious how to proceed, we might start on the other side, and see if the path is more apparent.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image343.gif                                Rewriting the tangents using the tangent identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image345.gif                           Multiplying the top and bottom by cos(a)cos(b)

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image347.gif  Distriuting and simplifying

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image349.gif         From above, we recognize this

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image351.gif                                                 Establishing the identity

These identities can also be used to solve equations.

Example 4

Solve File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image353.gif.

SOLUTION

By recognizing the left side of the equation as the result of the difference of angles identity for cosine, we can simplify the equation

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image353.gif           Apply the difference of angles identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image356.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image358.gif                                              Use the negative angle identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image360.gif

Since this is a special cosine value we recognize from the unit circle, we can quickly write the answers:

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image362.gif, where k is an integer

Combining Waves of Equal Period

A sinusoidal function of the form File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image364.gif can be rewritten using the sum of angles identity.

Example 5

Rewrite File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image366.gif as a sum of sine and cosine.

SOLUTION

Using the sum of angles identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image368.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image370.gif        Evaluate the sine and cosine

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image372.gif                    Distribute and simplify

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image374.gif

Notice that the result is a stretch of the sine added to a different stretch of the cosine, but both have the same horizontal compression, which results in the same period.

We might ask now whether this process can be reversed – can a combination of a sine and cosine of the same period be written as a single sinusoidal function?  To explore this, we will look in general at the procedure used in the example above.

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image364.gif                                     Use the sum of angles identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image376.gif                        Distribute the A

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image378.gif           Rearrange the terms a bit

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image380.gif

Based on this result, if we have an expression of the form File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image382.gif, we could rewrite it as a single sinusoidal function if we can find values A and C so that

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image382.gifFile:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image380.gif, which will require that:

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image385.gif  which can be rewritten as   File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image387.gif

To find A,

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image389.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image391.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image393.gif                    Apply the Pythagorean Identity and simplify

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image395.gif

Rewriting a Sum of Sine and Cosine as a Single Sine

To rewrite File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image382.gif as File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image397.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image399.gif, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image401.gif, and File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image403.gif

We can use either of the last two equations to solve for possible values of C. Since there will usually be two possible solutions, we will need to look at both to determine which quadrant C is in and determine which solution for C satisfies both equations.

Example 6

Rewrite File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image405.gif as a single sinusoidal function.

SOLUTION

Using the formulas above, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image407.gif, so A = 8. 

Solving for C,

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image409.gif, so File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image411.gif or File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image413.gif

However, since File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image415.gif, the angle that works for both is File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image413.gif

 

Combining these results gives us the expression

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image417.gif

Try it Now: 3

Rewrite File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image419.gif as a single sinusoidal function.

Rewriting a combination of sine and cosine of equal periods as a single sinusoidal function provides an approach for solving some equations.

Example 7

Solve File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image421.gif to find two positive solutions.

SOLUTION

To approach this, since the sine and cosine have the same period, we can rewrite them as a single sinusoidal function. 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image423.gif, so A = 5

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image425.gif, so File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image427.gif or File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image429.gif

Since File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image431.gif, a positive value, we need the angle in the first quadrant, C = 0.927.

Using this, our equation becomes

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image433.gif                         Divide by 5

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image435.gif                          Make the substitution u = 2x + 0.927

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image437.gif                                        The inverse gives a first solution

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image439.gif                       By symmetry, the second solution is

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image441.gif                      A third solution is

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image443.gif                               

Undoing the substitution, we can find two positive solutions for x.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image445.gif   or         File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image447.gif     or         File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image449.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image451.gif                         File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image453.gif                              File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image455.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image457.gif                           File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image459.gif                                File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image461.gif

Since the first of these is negative, we eliminate it and keep the two positive solutions, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image459.gif and File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image461.gif.

The Product-to-Sum and Sum-to-Product Identities

Identities

The Product-to-Sum Identities

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image464.gif

We will prove the first of these, using the sum and difference of angles identities from the beginning of the section.  The proofs of the other two identities are similar and are left as an exercise.

Proof of Product-to-Sum Identity for \(\sin(\alpha) \cos(\beta)\)

Recall the sum and difference of angles identities from earlier

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image282.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image284.gif

Adding these two equations, we obtain

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image466.gif

Dividing by 2, we establish the identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image468.gif

Example 8

Write File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image470.gif as a sum or difference.

SOLUTION

Using the product-to-sum identity for a product of sines

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image472.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image474.gif                               If desired, apply the negative angle identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image476.gif                                 Distribute

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image478.gif

Try it Now: 4

Evaluate \(\cos \left( \dfrac{11\pi}{12}  \cos \left( \dfrac{\pi}{12} \right)\).

Identities

The Sum-to-Product Identities

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image482.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image484.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image486.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image488.gif

We will again prove one of these and leave the rest as an exercise. 

Proof of Sum-to-product Identity for Sine Function

We begin with the product-to-sum identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image468.gif

We define two new variables:

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image491.gif

Adding these equations yields File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image493.gif, giving File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image495.gif

Subtracting the equations yields File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image497.gif, or File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image499.gif

Substituting these expressions into the product-to-sum identity above,

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image501.gif

Multiply by 2 on both sides

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image503.gif

Establishing the identity

Example 9

Evaluate File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image505.gif.

SOLUTION

Using the sum-to-product identity for the difference of cosines,

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image505.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image507.gif                         Simplify

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image509.gif                                            Evaluate

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image511.gif

Example 10

Prove the identity File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image513.gif.

SOLUTION

Since the left side seems more complicated, we can start there and simplify.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image515.gif                             Using the sum-to-product identities

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image517.gif       Simplify

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image519.gif                            Simplify further

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image521.gif                                         Rewrite as a tangent

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image523.gif                                        

Establishing the identity

Try it Now: 5

Notice that, using the negative angle identity, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image525.gif.  Use this along with the sum of sines identity to prove the sum-to-product identity for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image527.gif.

Example 11

Solve File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image529.gif for all solutions with File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image531.gif.

SOLUTION

In an equation like this, it is not immediately obvious how to proceed. One option would be to combine the two sine functions on the left side of the equation.  Another would be to move the cosine to the left side of the equation, and combine it with one of the sines.  For no particularly good reason, we’ll begin by combining the sines on the left side of the equation and see how things work out.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image529.gif                       Apply the sum to product identity on the left

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image534.gif   Simplify

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image536.gif                    Apply the negative angle identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image538.gif                      Rearrange the equation to be 0 on one side

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image540.gif                 Factor out the cosine

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image542.gif                        

Using the Zero Product Theorem we know that at least one of the two factors must be zero.  The first factor, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image544.gif, has period File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image546.gif, so the solution interval of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image531.gif represents one full cycle of this function.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image548.gif                                                  Substitute File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image550.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image552.gif                                                    On one cycle, this has solutions

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image554.gif or File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image556.gif                                         Undo the substitution

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image558.gif, so File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image560.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image562.gif, so File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image564.gif

The second factor, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image566.gif, has period of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image568.gif, so the solution interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image531.gif contains two complete cycles of this function.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image570.gif                                         Isolate the sine

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image572.gif                                                File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image574.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image576.gif                                                    On one cycle, this has solutions

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image578.gif or File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image580.gif                                         On the second cycle, the solutions are

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image582.gif or File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image584.gif  Undo the substitution

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image586.gif, so File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image588.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image590.gif, so File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image592.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image596.gif, so File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image598.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image600.gif, so File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image602.gif

Altogether, we found six solutions on File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image531.gif, which we can confirm by looking at the graph.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image604.gif

Important Topics of This Section

  • The sum and difference identities
  • Combining waves of equal periods
  • Product-to-sum identities
  • Sum-to-product identities
  • Completing proofs

 

Try it Now Answers

  1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image606.gif
  2. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image608.gif
  3. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image610.gif
  4. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image612.gif
  5. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image614.gif                                       Use negative angle identity for sine

      File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image616.gif                                      Use sum-to-product identity for sine

      File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image618.gif              Eliminate the parenthesis

      File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image620.gif                         Establishing the identity

Section 7.2 Exercises

 

Find an exact value for each of the following.

1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image622.gif                 2.  File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image624.gif              3. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image626.gif                4. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image628.gif              

5. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image630.gif                6. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image632.gif                 7. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image634.gif                8. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image636.gif

 

Rewrite in terms of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image638.gif and File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image640.gif.

9. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image642.gif         10. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image644.gif         11. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image646.gif        12. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image648.gif

 

Simplify each expression.

13. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image650.gif           14. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image652.gif          15. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image654.gif           16. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image656.gif

 

Rewrite the product as a sum.

17. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image658.gif                                 18. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image660.gif

19. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image662.gif                                     20. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image664.gif

 

Rewrite the sum as a product.

21. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image666.gif                                     22. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image668.gif

23. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image670.gif                                     24. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image672.gif

 

25. Given File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image674.gif and File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image676.gif, with a and b both in the interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image678.gif:

            a. Find File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image680.gif                               b. Find File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image682.gif

 

26. Given File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image684.gif and File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image686.gif, with a and b both in the interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image688.gif:

            a. Find File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image690.gif                               b. Find File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image692.gif

 

Solve each equation for all solutions.

27. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image694.gif

28. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image696.gif

29. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image698.gif

30. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image700.gif

Solve each equation for all solutions.

31. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image702.gif

32. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image704.gif

33. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image706.gif

34. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image708.gif

 

 

Rewrite as a single function of the form File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image710.gif.

35. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image712.gif                                   36. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image714.gif

37. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image716.gif                               38. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image718.gif

 

Solve for the first two positive solutions.

39. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image720.gif                            40. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image722.gif

41. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image724.gif                          42. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image726.gif

 

Simplify.

43. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image728.gif                                                44. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image730.gif

 

Prove the identity.

44. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image732.gif

45. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image734.gif

46. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image736.gif

47. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image738.gif

48. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image740.gif

49. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image742.gif

50. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image744.gif

51. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image746.gif

52. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image748.gif

Contributors

  • David Lippman (Pierce College)
  • Melonie Rasmussen (Pierce College)