# 8.6: Inverse Trigonometric Ratios

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Use the inverse trigonometric ratios to find an angle in a right triangle.
• Solve a right triangle.

## Review Queue

Find the lengths of the missing sides. Round your answer to the nearest tenth.

1. Draw an isosceles right triangle with legs of length 3. What is the hypotenuse?
2. Use the triangle from #3, to find the sine, cosine, and tangent of \begin{align*}45^\circ\end{align*}.

Know What? The longest escalator in North America is at the Wheaton Metro Station in Maryland. It is 230 feet long and is 115 ft. high. What is the angle of elevation, \begin{align*}x^\circ\end{align*}, of this escalator?

## Inverse Trigonometric Ratios

In mathematics, the word inverse means “undo.” For example, addition and subtraction are inverses of each other because one undoes the other. When we apply inverses to the trigonometric ratios, we can find acute angle measures as long as we are given two sides.

Inverse Tangent: Labeled \begin{align*}\tan^{-1}\end{align*}, the “-1” means inverse.

\begin{align*}\tan^{-1} \left (\frac{b}{a} \right ) = m \angle B && \tan^{-1} \left( \frac{a}{b} \right) = m \angle A\end{align*}

Inverse Sine: Labeled \begin{align*}\sin^{-1}\end{align*}.

\begin{align*}\sin^{-1} \left( \frac{b}{c} \right) = m \angle B && \sin^{-1} \left(\frac{a}{c} \right) = m \angle A\end{align*}

Inverse Cosine: Labeled \begin{align*}\cos^{-1}\end{align*}.

\begin{align*}\cos^{-1} \left(\frac{a}{c} \right) = m \angle B && \cos^{-1} \left( \frac{b}{c} \right) = m \angle A\end{align*}

In order to find the measure of the angles, you will need you use your calculator. On most scientific and graphing calculators, the buttons look like \begin{align*}[ \text{SIN}^{-1}], [ \text{COS}^{-1}]\end{align*}, and \begin{align*}[\text{TAN}^{-1}]\end{align*}. You might also have to hit a shift or \begin{align*}2^{nd}\end{align*} button to access these functions.

Example 1: Use the sides of the triangle and your calculator to find the value of \begin{align*}\angle A\end{align*}. Round your answer to the nearest tenth of a degree.

Solution: In reference to \begin{align*}\angle A\end{align*}, we are given the opposite leg and the adjacent leg. This means we should use the tangent ratio.

\begin{align*}\tan A = \frac{20}{25} = \frac{4}{5}\end{align*}. So, \begin{align*}\tan^{-1} \frac{4}{5} = m \angle A\end{align*}. Now, use your calculator.

If you are using a TI-83 or 84, the keystrokes would be: \begin{align*}[2^{nd}]\end{align*}[TAN]\begin{align*}\left( \frac{4}{5} \right)\end{align*} [ENTER] and the screen looks like:

\begin{align*}m \angle A = 38.7^\circ\end{align*}

Example 2: \begin{align*}\angle A\end{align*} is an acute angle in a right triangle. Find \begin{align*}m \angle A\end{align*} to the nearest tenth of a degree.

a) \begin{align*}\sin A = 0.68\end{align*}

b) \begin{align*}\cos A = 0.85\end{align*}

c) \begin{align*}\tan A = 0.34\end{align*}

Solution:

a) \begin{align*}m \angle A = \sin^{-1} 0.68=42.8^\circ\end{align*}

b) \begin{align*}m \angle A = \cos^{-1} 0.85=31.8^\circ\end{align*}

c) \begin{align*}m \angle A = \tan^{-1} 0.34=18.8^\circ\end{align*}

## Solving Triangles

To solve a right triangle, you need to find all sides and angles in a right triangle, using sine, cosine or tangent, inverse sine, inverse cosine, or inverse tangent, or the Pythagorean Theorem.

Example 3: Solve the right triangle.

Solution: To solve this right triangle, we need to find \begin{align*}AB, m \angle C\end{align*} and \begin{align*}m \angle B\end{align*}. Only use the values you are given.

\begin{align*}\underline{AB}\end{align*}: Use the Pythagorean Theorem.

\begin{align*}24^2 + AB^2 &= 30^2\\ 576 + AB^2 &= 900\\ AB^2 &= 324\\ AB &= \sqrt{324} = 18\end{align*}

\begin{align*}\underline{m \angle B}\end{align*}: Use the inverse sine ratio.

\begin{align*}\sin B &= \frac{24}{30} = \frac{4}{5}\\ \sin^{-1} \left( \frac{4}{5} \right) &= 53.1^\circ = m \angle B\end{align*}

\begin{align*}\underline{m \angle C}\end{align*}: Use the inverse cosine ratio.

\begin{align*}\cos C = \frac{24}{30} = \frac{4}{5} \longrightarrow \cos^{-1}\left( \frac{4}{5} \right) = 36.9^\circ = m \angle C\end{align*}

Example 4: Solve the right triangle.

Solution: To solve this right triangle, we need to find \begin{align*}AB, BC\end{align*} and \begin{align*}m \angle A\end{align*}.

\begin{align*}\underline{AB}\end{align*}: Use sine ratio.

\begin{align*}\sin 62^\circ &= \frac{25}{AB}\\ AB &= \frac{25}{\sin 62^\circ}\\ AB & \approx 28.31\end{align*}

\begin{align*}\underline{BC}\end{align*}: Use tangent ratio.

\begin{align*}\tan 62^\circ &= \frac{25}{BC}\\ BC &= \frac{25}{\tan 62^\circ}\\ BC & \approx 13.30\end{align*}

\begin{align*}\underline{m \angle A}\end{align*}: Use Triangle Sum Theorem

\begin{align*}62^\circ + 90^\circ + m \angle A &= 180^\circ\\ m \angle A &= 28^\circ\end{align*}

Example 5: Solve the right triangle.

Solution: The two acute angles are congruent, making them both \begin{align*}45^\circ\end{align*}. This is a 45-45-90 triangle. You can use the trigonometric ratios or the special right triangle ratios.

Trigonometric Ratios

\begin{align*}\tan 45^\circ &= \frac{15}{BC} && \sin 45^\circ = \frac{15}{AC}\\ BC &= \frac{15}{\tan 45^\circ} = 15 && \quad \ AC = \frac{15}{\sin 45^\circ} \approx 21.21\end{align*}

45-45-90 Triangle Ratios

\begin{align*}BC = AB = 15, AC = 15 \sqrt{2} \approx 21.21\end{align*}

## Real-Life Situations

Example 6: A 25 foot tall flagpole casts a 42 feet shadow. What is the angle that the sun hits the flagpole?

Solution: Draw a picture. The angle that the sun hits the flagpole is \begin{align*}x^\circ\end{align*}. We need to use the inverse tangent ratio.

\begin{align*}\tan x &= \frac{42}{25}\\ \tan^{-1} \frac{42}{25} & \approx 59.2^\circ = x\end{align*}

Example 7: Elise is standing on top of a 50 foot building and sees her friend, Molly. If Molly is 35 feet away from the base of the building, what is the angle of depression from Elise to Molly? Elise’s eye height is 4.5 feet.

Solution: Because of parallel lines, the angle of depression is equal to the angle at Molly, or \begin{align*}x^\circ\end{align*}. We can use the inverse tangent ratio.

\begin{align*}\tan^{-1} \left( \frac{54.5}{30} \right) = 61.2^\circ = x\end{align*}

Know What? Revisited To find the escalator’s angle of elevation, use the inverse sine.

\begin{align*}\sin^{-1} \left( \frac{115}{230} \right)= 30^\circ \qquad \quad \text{The angle of elevation is} \ 30^\circ.\end{align*}

## Review Questions

• Questions 1-6 are similar to Example 1.
• Questions 7-12 are similar to Example 2.
• Questions13-21 are similar to Examples 3 and 4.
• Questions 22-24 are similar to Examples 6 and 7.
• Questions 25-30 are a review of the trigonometric ratios.

Use your calculator to find \begin{align*}m \angle A\end{align*} to the nearest tenth of a degree.

Let \begin{align*}\angle A\end{align*} be an acute angle in a right triangle. Find \begin{align*}m \angle A\end{align*} to the nearest tenth of a degree.

1. \begin{align*}\sin A = 0.5684\end{align*}
2. \begin{align*}\cos A = 0.1234\end{align*}
3. \begin{align*}\tan A = 2.78\end{align*}
4. \begin{align*}\cos^{-1} 0.9845\end{align*}
5. \begin{align*}\tan^{-1} 15.93\end{align*}
6. \begin{align*}\sin^{-1} 0.7851\end{align*}

Solving the following right triangles. Find all missing sides and angles. Round any decimal answers to the nearest tenth.

Real-Life Situations Use what you know about right triangles to solve for the missing angle. If needed, draw a picture. Round all answers to the nearest tenth of a degree.

1. A 75 foot building casts an 82 foot shadow. What is the angle that the sun hits the building?
2. Over 2 miles (horizontal), a road rises 300 feet (vertical). What is the angle of elevation?
3. A boat is sailing and spots a shipwreck 650 feet below the water. A diver jumps from the boat and swims 935 feet to reach the wreck. What is the angle of depression from the boat to the shipwreck?

Examining Patterns Below is a table that shows the sine, cosine, and tangent values for eight different angle measures. Answer the following questions.

\begin{align*}10^\circ\end{align*} \begin{align*}20^\circ\end{align*} \begin{align*}30^\circ\end{align*} \begin{align*}40^\circ\end{align*} \begin{align*}50^\circ\end{align*} \begin{align*}60^\circ\end{align*} \begin{align*}70^\circ\end{align*} \begin{align*}80^\circ\end{align*}
Sine 0.1736 0.3420 0.5 0.6428 0.7660 0.8660 0.9397 0.9848
Cosine 0.9848 0.9397 0.8660 0.7660 0.6428 0.5 0.3420 0.1736
Tangent 0.1763 0.3640 0.5774 0.8391 1.1918 1.7321 2.7475 5.6713
1. What value is equal to \begin{align*}\sin 40^\circ\end{align*}?
2. What value is equal to \begin{align*}\cos 70^\circ\end{align*}?
3. Describe what happens to the sine values as the angle measures increase.
4. Describe what happens to the cosine values as the angle measures increase.
5. What two numbers are the sine and cosine values between?
6. Find \begin{align*}\tan85^\circ, \tan 89^\circ\end{align*}, and \begin{align*}\tan 89.5^\circ\end{align*} using your calculator. Now, describe what happens to the tangent values as the angle measures increase.

## Review Queue Answers

1. \begin{align*}\sin 36^\circ = \frac{y}{7} \qquad \cos 36^\circ = \frac{x}{7}\!\\ {\;} \qquad y = 4.11 \qquad \quad \ \ x = 5.66\end{align*}
2. \begin{align*}\cos 12.7^\circ = \frac{40}{x} \qquad \ \tan 12.7^\circ = \frac{y}{40}\!\\ {\;} \qquad \ \ \ x = 41.00 \qquad \qquad \ \ y = 9.01\end{align*}
3. \begin{align*}{\;} \ \sin 45^\circ = \frac{3}{3 \sqrt{2}} = \frac{\sqrt{2}}{2}\!\\ {\;} \ \cos 45^\circ = \frac{3}{3 \sqrt{2}} = \frac{\sqrt{2}}{2}\!\\ {\;} \ \tan 45^\circ = \frac{3}{3} = 1\end{align*}

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