6.2: Technology and Right Triangle Trigonometry

Note to the Instructor (click to expand) We finally reach the moment where we are evaluating trigonometric functions at non-special acute angles. As such, technology (e.g., Desmos or a calculator) will be necessary. This will also be needed to find the acute angle within a right triangle given the sides of the triangle. The following is a list of learning objectives for this section. Learning Objectives (click to expand) Use a calculator to approximate the value of a trigonometric function. Use a calculator to approximate an acute angle given the value of a trigonometric function.

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Using a Calculator to Compute the Values of Trigonometric Functions

The trigonometric values for non-special angles were originally collected into tables and are available on scientific calculators (and other, more recent, technology). For example, let's find the sine of \(50^{\circ}\).

Note: Change to Degree Mode

Before evaluating any trigonometric function, ensure your technology is set to the correct "mode." In Trigonometry, the two standard modes are degree mode and radian mode. At this point, we will only be working with angles measured in degrees, so you must only concern yourself with degree mode. If using a scientific or graphing calculator, refer to your instruction manual on setting the calculator to degree mode. The scientific calculator in Desmos is set to degree mode by default, so you do not have to make any adjustments if using that technology; however, the graphing calculator in Desmos defaults to radian mode. To change Desmos' graphing calculator to degree mode, open the "Graph Settings" and select "Degrees."

Example \( \PageIndex{ 1 } \)

Approximate \( \sin\left( 50^{ \circ } \right) \).

Scientific and Graphing Calculators

Press the \( \mathrm{SIN} \) button. Your calculator will display\[\sin (\nonumber \]with an open parenthesis as the prompt to enter an angle. Type \(50\) and press \( \mathrm{ENTER} \) (since your calculator should already be in degree mode, it will interpret the \( 50 \) as \( 50^{ \circ } \)). The calculator returns \(0.7660444431\) (please verify).

Desmos

Using the scientific calculator in Desmos, type \( \mathrm{sin(} 50 \mathrm{)} \) and hit \( \mathrm{ENTER} \) (since the scientific calculator in Desmos defaults to degree mode, it will interpret the \( 50 \) as \( 50^{ \circ } \)). Desmos returns \(0.7660444431\) (please verify).

It is important to note that the values returned by the technology used in Example \( \PageIndex{ 1 } \) are not exact values; it turns out that the sine of \(50^{\circ}\) is an irrational number, and your calculator shows as many digits as its display will allow (not all sine values are as "nice" as the sine of \(30^{\circ}\)). Unless otherwise instructed, we round to four decimal places, so we write\[\sin \left(50^{\circ}\right) \approx 0.7660.\nonumber\]Take note of using the approximation sign, \( \approx \), instead of the equal sign - this is important!

Checkpoint \( \PageIndex{ 1 } \)

1. Use a calculator to complete the table, rounding answers to four decimal places.

   \(\theta\)	\(0^{\circ}\)	\(10^{\circ}\)	\(20^{\circ}\)	\(30^{\circ}\)	\(40^{\circ}\)	\(50^{\circ}\)	\(60^{\circ}\)	\(70^{\circ}\)	\(80^{\circ}\)	\(89^{\circ}\)
   \(\tan\left( \theta\right)\)

2. What do you notice about the values of \(\tan\left( \theta\right)\) as \(\theta\) increases from \(0^{\circ}\) to \(89^{\circ}\)? If the values of \(\tan \left(\theta\right)\) are plotted against the values of \(\theta\), will the graph be a straight line? Why or why not?

Answers

1. \(\theta\)	\(0^{\circ}\)	\(10^{\circ}\)	\(20^{\circ}\)	\(30^{\circ}\)	\(40^{\circ}\)	\(50^{\circ}\)	\(60^{\circ}\)	\(70^{\circ}\)	\(80^{\circ}\)	\(90^{\circ}\)
   \(\tan \left(\theta\right)\)	0	0.1763	0.3640	0.5774	0.8391	1.1918	1.7321	2.7475	5.6713	57.2900

2. The values of \(\tan\left( \theta\right)\) increase seemingly without bound as \(\theta\) increases from \(0^{\circ}\) to \(89^{\circ}\). The graph will not be a line because the slopes between successive points are not constant.

Example \(\PageIndex{2}\)

Use technology to compute the value of each trigonometric function.

1. \( \cos\left( 23.7^{ \circ } \right) \)
2. \( \tan\left( 72.13^{ \circ } \right) \)
3. \( \sin^2\left( 10^{ \circ } \right) \)

Solutions

1. \( \cos\left( 23.7^{ \circ } \right) \approx 0.9156625933 \approx 0.9157 \)
2. \( \tan\left( 72.13^{ \circ } \right) \approx 3.101611189 \approx 3.1016 \)
3. To compute \( \sin^2\left( 10^{ \circ } \right) \), we need to recall that\[ \sin^2\left( 10^{ \circ } \right) = \left[ \sin\left( 10^{ \circ } \right) \right]^2. \nonumber \]Therefore, we start by getting the approximation \( \sin\left( 10^{ \circ } \right) \approx 0.1736481777 \). Using as many of these digits as possible, we then square the result to get\[ \sin^2\left( 10^{ \circ } \right) \approx \left( 0.1736481777 \right)^2 \approx 0.03015368961 \approx 0.0304. \nonumber \]

Caution: Save Rounding for the Final Answer

Example \( \PageIndex{ 2c } \) demonstrates a significant point that bears repeating - rounding is for display purposes only! Never round a result and then use that rounded result for other computations.

For example, had we rounded our approximation of \( \sin\left( 10^{ \circ } \right) \) to be 0.1736 and then squared, we would have incorrectly stated \( \sin^2\left( 10^{ \circ } \right) \approx 0.0301 \). While the difference between this result and the more accurate result of \( 0.0304 \) may seem insignificant, these rounding errors can propagate in such a way as to make massive differences between answers.

Checkpoint \( \PageIndex{ 2 } \)

1. Use a calculator to complete the table. Round the values of sine and cosine to four decimal places.

   \(\theta\)	\(0^{\circ}\)	\(10^{\circ}\)	\(20^{\circ}\)	\(30^{\circ}\)	\(40^{\circ}\)	\(50^{\circ}\)	\(60^{\circ}\)	\(70^{\circ}\)	\(80^{\circ}\)	\(90^{\circ}\)
   \(\sin\left( \theta\right)\)
   \(\cos\left( \theta\right)\)

2. Is there a pattern to the values of sine and cosine? Why this is true? (Hint: If one (non-right) angle of a right triangle measures \(x\) degrees, how big is the other angle? Sketch the triangle and label each angle's opposite and adjacent sides.)

Answers

1. \(\theta\)	\(0^{\circ}\)	\(10^{\circ}\)	\(20^{\circ}\)	\(30^{\circ}\)	\(40^{\circ}\)	\(50^{\circ}\)	\(60^{\circ}\)	\(70^{\circ}\)	\(80^{\circ}\)	\(90^{\circ}\)
   \(\sin \left(\theta\right)\)	0	0.1737	0.3420	0.5	0.6428	0.7660	0.8660	0.9397	0.9848	1
   \(\cos \left(\theta\right)\)	1	0.9849	0.9397	0.8660	0.7660	0.6428	0.5	0.3420	0.1737	0

2. The cosine of \(\theta\) is equal to the sine of the complement of \(\theta\), or \(\cos \left(\theta\right) = \sin(90 − \theta)\). This result confirms the Cofunction Identity from the previous section.

Now that we have access to technology to approximate the value of trigonometric functions at non-special angles, we can revisit simplifying and evaluating trigonometric expressions.

Example \( \PageIndex{ 3} \)

Simplify, and evaluate for \(z=40^{\circ}\).\[3 \sin \left(z\right)-\sin \left(z\right) \tan \left(z\right)+3 \sin \left(z\right)\nonumber \]

Solution

We can combine like terms to get\[6 \sin \left(z\right)-\sin \left(z\right) \tan \left(z\right)\nonumber \]Because \(40^{\circ}\) is not one of the angles for which we know exact trig values, we use a calculator to evaluate the expression. It is very important to make sure we save any rounding until the very end of the computation. Therefore, we enter the following keystrokes into our calculator (or other technology):\[ 6 \quad \times \quad \mathrm{SIN} \quad 40 \quad \mathrm{)} \quad \mathrm{-} \quad \mathrm{SIN} \quad 40 \quad \mathrm{)} \quad \times \quad \mathrm{TAN} \quad 40 \quad \mathrm{ENTER} \nonumber \]Doing so, we get approximately 3.3174.

Checkpoint \( \PageIndex{ 3 } \)

Simplify, and evaluate for \(x=25^{\circ}, y=70^{\circ}\).\[3 \cos \left(x\right)+\cos \left(y\right)-2 \cos \left(y\right)+\cos \left(x\right)\nonumber \]

Answer

\(3.2832\)

Unlike the fundamental trigonometric functions, calculators do not have keys for the cosecant, secant, and cotangent functions. Instead, we calculate their values as reciprocals.

Example \( \PageIndex{ 4 } \)

Use technology to approximate \(\sec \left(47^{\circ}\right)\) to three decimal places.

Solution

Enter\[1 \div \cos 47 ) \text { ENTER } \nonumber \]to obtain \(\sec \left(47^{\circ}\right) \approx 1.466\). Or we can calculate \(\cos \left(47^{\circ}\right)\) first, and then use the reciprocal key:\[\cos 47 ) \text{ ENTER } x^{-1} \text{ ENTER } \nonumber \]

Checkpoint \( \PageIndex{ 4 } \)

Use technology to approximate \(\csc \left(82^{\circ}\right)\) to three decimal places.

Answer

1.010

Finding an Acute Angle Using a Calculator

While watching her niece at the playground, Nhat wonders how steep the slide is. She takes out her measuring tape and finds the slide to be 77 inches high over a horizontal distance of 136 inches, as shown below.

Figure \( \PageIndex{ 1 } \)

Nhat knows that one way to describe the steepness of an incline is to calculate its slope, which, in this case, is\[\dfrac{\Delta y}{\Delta x} = \dfrac{77}{136} \approx 0.5662.\nonumber \]However, she wants to know what angle the slide makes with the horizontal. Thinking about it for a moment, she realizes that the slope she had just calculated is also the tangent of the angle she wants (can you see this from the figures above?). This begs the question: If we know the tangent of an angle, can we find the angle? The answer is yes!

Locate the key labeled \(\mathrm{TAN}^{−1}\) on your calculator; it is probably the second function above the \(\mathrm{TAN}\) key. Enter\[ \mathrm{2nd} \quad \mathrm{TAN} \quad 0.5662 \nonumber \]to find that\[\tan ^{-1} \left( 0.5662 \right) \approx 29.5185^{\circ}.\nonumber \]This means that \(29.5185^{\circ}\) is the approximate angle whose tangent is \(0.5662\). We read the notation, \( \tan^{-1}\left( 0.5662 \right) \approx 29.5185^{\circ} \) as "inverse tangent of \(0.5662\) is approximately \(29.5185\) degrees."

Note: Use Degree Mode

Until later in this course, you should make sure your calculator is set to "degree" mode. If you do not do this, your answers, while correct in a different angular measure system, will be incorrect.

Caution: Do Not Use Rounded Values for Calculations

Before describing the inverse tangent, we must note the flaw in our work. We approximated the value of the fraction, \( \frac{77}{136} \), and then used that approximation to approximate the angle - this is very bad form. In truth, we should not have approximated \( \frac{77}{136} \), but instead just computed\[ \tan^{-1}\left( \dfrac{77}{136} \right) \approx 29.5175^{ \circ }. \nonumber \]Do you see that the returned angle from the inverse tangent is slightly different from our calculations using the rounded approximation of the fraction?

The moral of the story is that, whenever possible, do not use rounded or approximate values to get yet another approximation - this leads to a propagation of rounding errors.

When we find \(\tan ^{-1}\) of a number, we find an angle whose tangent is that number. Similarly, \(\sin ^{-1}\) and \(\cos ^{-1}\) are read as "inverse sine" and "inverse cosine." They find an angle with the given sine or cosine.

Example \( \PageIndex{ 5} \)

Find the angle whose sine is \(0.6834\).

Solution

Enter \[ \mathrm{2nd} \quad \mathrm{SIN} \quad 0.6834\nonumber \]into your calculator to find\[\sin ^{-1} \left(0.6834\right) \approx 43.11^{\circ}. \nonumber \]So \(43.11^{\circ}\) is the angle whose sine is approximately \(0.6834\). Alternatively, we can say that\[\sin \left(43.11^{\circ}\right) \approx 0.6834. \nonumber \]You can check the last equation on your calculator.

Note: Equivalent Statements

In Example \( \PageIndex{ 5} \), the two equations\[\sin \left(43.11^{\circ} \right) \approx 0.6834 \quad \text { and } \quad \sin ^{-1}\left( 0.6834\right) \approx 43.11^{\circ}\nonumber \]say the same thing in different ways.

Caution: Inverse Trigonometric Functions are NOT Reciprocals

The notation \(\sin ^{-1} \left(x\right)\) does not mean \(\frac{1}{\sin\left( x\right)}\)!

We indeed use negative exponents to indicate reciprocals of numbers, for example, \(a^{-1}=\frac{1}{a}\) and \(3^{-1}=\frac{1}{3}\). However, "sin" by itself is not a variable.

• \(\sin ^{-1} \left(x\right)\) means "the angle whose sine is \(x\)"
• \(\frac{1}{\sin \left(x\right)}\) means "the reciprocal of the sine of angle \(x\)"

(You may recall that \(f^{-1}(x)\) denotes the inverse function for \(f(x)\). We will study inverse trigonometric functions in a later chapter.)

Checkpoint \( \PageIndex{ 5} \)

Write the following fact in two different ways: \(68^{\circ}\) is the angle whose cosine is \(0.3746\).

Answer

\(\cos \left(68^{\circ}\right)=0.3746\) or \(\cos ^{-1}\left(0.3746\right)=68^{\circ}\)

Note: Inputs and Outputs of Inverse Trigonometric Functions

Without going into too much detail (yet), it is essential to note that an inverse trigonometric function's input (a.k.a., the argument) is a ratio. The output is an angle.

Example \(\PageIndex{6}\)

Find the acute angle for which the equation is true. Round answers to the nearest tenth.

1. \( \cos\left( \beta \right) = 0.28 \)
2. \( \tan\left( \kappa \right) = 1.93 \)

Solutions

1. We use the following button sequence:\[ \mathrm{2nd} \quad \mathrm{COS} \quad 0.28 \quad \mathrm{ENTER}. \nonumber \]This returns \( 73.73979529 \). Hence, \( \beta \approx 73.7^{ \circ } \).
2. Using the sequence\[ \mathrm{2nd} \quad \mathrm{TAN} \quad 1.93 \quad \mathrm{ENTER}, \nonumber \]we get \( 62.60975803 \). Hence, \( \kappa \approx 62.6^{ \circ } \).

Checkpoint \(\PageIndex{6}\)

Find the acute angle for which \( \tan\left( \theta \right) = 1.23 \). Round your answer to the nearest tenth.

Answer

\( \theta \approx 50.9^{ \circ } \)

Finding acute angles is slightly more complex when the equation involves the cosecant, secant, or cotangent. A good rule of thumb is to rewrite the equation in terms of the fundamental trigonometric functions using the Reciprocal Identities.

Example \(\PageIndex{7}\)

Suppose \( x \) is an acute angle within a right triangle. Solve each equation. Round answers to two decimal places.

1. \( \sec\left( x \right) = 8.104 \)
2. \( 3 \cot\left( x \right) - 2 = 7 \)

Solutions

1. We first rewrite the equation as\[\dfrac{1}{\cos\left( x \right)} = 8.104 \implies \cos\left( x \right) = \dfrac{1}{8.104} \nonumber \]We then enter\[ \mathrm{2nd} \quad \mathrm{COS} \quad 1 \quad \div \quad 8.104\nonumber \]into the calculator to find\[\cos ^{-1} \left( \frac{1}{8.104}\right) \approx 82.91^{\circ}. \nonumber \]Therefore, the solution to \( \sec\left( x \right) = 8.104 \) is \( x \approx 82.91^{\circ} \).
2. The equation handed to us needs to be simplified into the form \( \cot\left( \text{angle} \right) = \text{number} \) before we can find the angle.\[\begin{array}{rrcl}
   & 3 \cot\left( x \right) - 2 & = & 7 \\[6pt] \implies & 3 \cot \left( x \right) & = & 9 \\[6pt] \implies & \cot \left( x \right) & = & 3 \\[6pt] \end{array} \nonumber \]We now rewrite the cotangent as the tangent using the Reciprocal Identities.\[ \cot\left( x \right) = 3 \implies \dfrac{1}{\tan\left( x \right)} = 3 \implies \tan\left( x \right) = \dfrac{1}{3} \nonumber \]Finally, we grab our technology and enter\[ \mathrm{2nd} \quad \mathrm{TAN} \quad 1 \quad \div \quad 3\nonumber \]into the calculator to find\[\tan ^{-1} \left(\frac{1}{3}\right) \approx 18.43^{\circ} \nonumber \]Therefore, the solution to \( 3 \cot\left( x \right) - 2 = 7 \) is \( x \approx 18.43^{\circ} \).

Checkpoint \(\PageIndex{7}\)

Let \( \omega \) be an acute angle within a right triangle. Solve \( 2 \csc\left( \omega \right) + 3 = 11 \). Round the answer to two decimal places.

Answer

\(\omega \approx 14.48^{\circ}\)

Activities 

Activity: Trigonometric Ratios

1. Using Ratios and Proportions
   Recall that two related quantities or variables are proportional if their ratio is always the same.
   1. 1. On any given day, the cost of filling up your car’s gas tank is proportional to the number of gallons of gas you buy. For each purchase below, compute the ratio\[\dfrac{\text { total cost of gasoline }}{\text { number of gallons }} \nonumber\]

         Gallons of Gas Purchased	Total Cost	\(\dfrac{\text { Dollars }}{\text { Gallon }}\)
         5	$14.45
         12	$34.68
         18	$52.02

      2. Write an equation that you can solve to answer the question: How much does 21 gallons of gas cost? Use the ratio \(\frac{\text { Dollars }}{\text { Gallon }}\) in your equation.
      3. Write an equation that you can solve to answer the question: How many gallons of gas can you buy for \(\$ 46.24\)? Use the ratio \(\frac{\text { Dollars }}{\text { Gallon }}\) in your equation.
   2. A recipe for coffee cake calls for \(\frac{3}{4}\) cup of sugar and \(1 \frac{3}{4}\) cup of flour.
      1. What is the ratio of sugar to flour? Write your answer as a common fraction, then give a decimal approximation rounded to four places.
         
         
         For parts (b) and (c) below, write an equation that you can solve to answer the question. Use the ratio \(\frac{\text { Amount of sugar }}{\text { Amount of flour }}\) your equation.
      2. How much sugar should you use if you use 4 cups of flour? Compute your answer in two ways: write the ratio as a common fraction and then write the ratio as a decimal approximation. Are your answers the same?
      3. How much flour should you use if you use 4 cups of sugar? Compute your answer in two ways: write the ratio as a common fraction and then write the ratio as a decimal approximation. Are your answers the same?
   3. You are making a scale model of the Eiffel Tower, which is 324 meters tall and 125 meters wide at its base.

      Figure \( \PageIndex{ 2 } \)

      

      1. Compute the ratio of the base width to the tower's height. Round your answer to four decimal places.
         
         Use your ratio to write equations and answer the questions below:
      2. If the base of your model is 8 inches wide, how tall should the model be?
      3. If you make a larger model that is 5 feet tall, how wide will the base be?
2. Similar Triangles
   1. Recall that two triangles are similar if their corresponding sides are proportional. The corresponding angles of similar triangles are equal.
      1. What is the ratio of the two given sides in each triangle? Are the corresponding sides of the three triangles proportional? How do we know that \(\alpha = \beta = \gamma\)?

         Figure \( \PageIndex{ 3 } \)

         

      2. Find the hypotenuse of each right triangle.
      3. Use the sides of the appropriate triangle to compute \(\sin \alpha, \sin \beta\), and \(\sin \gamma\). Round your answers to four decimal places. Does the sine of an angle depend on the lengths of its sides?
      4. How do you know that the triangle below is similar to the three triangles in part (a)? Write an equation using the ratio from part (c) to find \(x\).

         Figure \( \PageIndex{ 4 } \)

         

   2. In the three right triangles below, the angle \(\theta\) is the same size.

      Figure \( \PageIndex{ 5 } \)

      

      1. Use the first triangle to calculate \(\cos \theta\). Round your answer to four decimal places.
      2. In the second triangle, explain why \(\frac{x}{4.3} = \frac{10}{13}\). Write an equation using your answer to part (a) and solve it to find \(x\).
      3. Write and solve an equation to find \(z\) in the third triangle.
   3. 1. Use your calculator to find the value of \(\frac{h}{2.4}\). (Hint: Which trigonometric ratio should you use?) What is the length of side \(h\)?

         Figure \( \PageIndex{ 6 } \)

         

      2. What is the value of \(\frac{6}{w}\) for the triangle below? Write an equation and solve for \(w\).

         Figure \( \PageIndex{ 7 } \)

         

      3. Write an equation and solve it to find \(x\) in the triangle above.

Activity: Obtuse Angles

An obtuse angle has measure between \(90^{\circ}\) and \(180^{\circ}\), exclusively. In this activity, we will investigate the trigonometric functions of an obtuse angle.

Figure \( \PageIndex{ 8 } \)

1. Draw an angle \(\theta\) in standard position with the point \(P(6,4)\) on its terminal side.
2. Find \(r\), the distance from the origin to \(P\).
3. Calculate all six trigonometric functions of \(\theta\). Give both exact answers and decimal approximations rounded to four places.
4. Use the inverse cosine key on your calculator to find \(\theta\). Use your calculator to verify the values of \(\sin \left(\theta\right), \cos \left(\theta\right)\), and \(\tan \left(\theta\right)\) that you found in part (c).
5. Draw another angle \(\phi\) in standard position with the point \(Q(-6,4)\) on its terminal side. Explain why \(\phi\) is the supplement of \(\theta\). (Hint: Consider the right triangles formed by drawing vertical lines from \(P\) and \(Q\)).
6. Can you use the right triangle definition of the trigonometric functions to compute the sine and cosine of \(\phi\)? Why or why not?
7. Calculate all six trigonometric functions of \( \phi \). How are these trigonometric values of \(\phi\) related to the trigonometric values of \(\theta\)?
8. Explain why \(\theta\) and \(\phi\) have the same sine but different cosines.
9. Use the inverse cosine key on your calculator to find \(\phi\). Use your calculator to verify the values of \(\sin \left(\phi\right), \cos \left(\phi\right)\), and \(\tan \left(\phi\right)\) that you found in part (g).
10. Compute \(180^{\circ}-\phi\). What answer should you expect to get?