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# Inverse Trigonometric Functions

## Solving for an angle given a trigonometric ratio.

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Inverse Trigonometric Functions

One day after school you are trying out for the track team. Your school has a flag pole at the very end of its football field. The pole is 50 feet tall. While standing at the end of the track, you know that the distance between you and the flag pole is 350 feet. Being curious about such things, you decide to find the angle between the ground and the top of the flag pole from where you are standing.

Can you solve this problem?

### Inverse Trigonometric Functions

Consider the right triangle below.

From this triangle, we know how to determine all six trigonometric functions for both \begin{align*}\angle{C}\end{align*} and \begin{align*}\angle{T}\end{align*}.

From any of these functions we can also find the value of the angle, using our graphing calculators. You might recall that \begin{align*}\sin 30^\circ = \frac{1}{2}\end{align*}. If you type 30 into your graphing calculator and then hit the SIN button, the calculator yields 0.5. (Make sure your calculator’s mode is set to degrees.)

Conversely, with the triangle above, we know the trig ratios, but not the angle. In this case the inverse of the trigonometric function must be used to determine the measure of the angle. These functions are located above the SIN, COS, and TAN buttons on the calculator. To access one of these functions, press \begin{align*}2^{nd}\end{align*} and the appropriate button and the measure of the angle appears on the screen.

\begin{align*}\cos T = \frac{24}{25} \rightarrow \cos^{-1} \frac{24}{25} = T\end{align*} from the calculator we get

For the problems below, use your calculator the find the angle measurement of the trig functions.

1. \begin{align*}\sin x = 0.687\end{align*}

Plug into calculator.

\begin{align*}\sin^{-1} 0.687 = 43.4^\circ\end{align*}

2. \begin{align*}\tan x = \frac{4}{3}\end{align*}

Plug into calculator.

\begin{align*}\tan^{-1} \frac{4}{3} = 53.13^\circ\end{align*}

#### Real-World Application: Conveyor Belt

You live on a farm and your chore is to move hay from the loft of the barn down to the stalls for the horses. The hay is very heavy and to move it manually down a ladder would take too much time and effort. You decide to devise a make shift conveyor belt made of bed sheets that you will attach to the door of the loft and anchor securely in the ground. If the door of the loft is 25 feet above the ground and you have 30 feet of sheeting, at what angle do you need to anchor the sheets to the ground?

From the picture, we need to use the inverse sine function.

\begin{align*}\sin \theta & = \frac{25 \ feet}{30 \ feet}\\ \sin \theta & = 0.8333\\ \sin^{-1} (\sin \theta) & = \sin ^{-1} 0.8333\\ \theta & = 56.4^\circ\end{align*}

The sheets should be anchored at an angle of \begin{align*}56.4^\circ\end{align*}.

### Examples

#### Example 1

Earlier, you were asked to find the angle between the ground and the top of the flag pole from where you are standing.

Using your knowledge of inverse trigonometric functions, you can set up a tangent relationship to solve for the angle:

\begin{align*} \tan \theta = \frac{50}{350}\\ \theta = \tan^{-1} \frac{50}{350}\\ \theta \approx 8.13^\circ\\ \end{align*}

#### Example 2

Find the angle measure for the trig function below.

\begin{align*}\sin x = 0.823\end{align*}

Plug into calculator.

\begin{align*}\sin^{-1} 0.823 \approx 55.39^\circ\end{align*}

#### Example 3

Find the angle measure for the trig function below.

\begin{align*}\cos x = -0.112\end{align*}

Plug into calculator.

\begin{align*}\cos^{-1} -0.112 \approx 96.43^\circ\end{align*}

#### Example 4

Find the angle measure for the trig function below.

\begin{align*}\tan x = 0.2\end{align*}

Plug into calculator.

\begin{align*}\tan^{-1} 0.2 \approx 11.31^\circ\end{align*}

### Review

Use inverse trigonometry to find the angle measure of angle A for each angle below.

1. \begin{align*}\sin A = 0.839\end{align*}
2. \begin{align*}\cos A = 0.19\end{align*}
3. \begin{align*}\tan A = 0.213\end{align*}
4. \begin{align*}\csc A = 1.556\end{align*}
5. \begin{align*}\sec A = 2.063\end{align*}
6. \begin{align*}\cot A = 2.356\end{align*}
7. \begin{align*}\csc A = 8.206\end{align*}
8. \begin{align*}\sin A = 0.9994\end{align*}
9. \begin{align*}\cot A = 1.072\end{align*}
10. \begin{align*}\cos A = 0.174\end{align*}
11. \begin{align*}\tan A = 1.428\end{align*}
12. \begin{align*}\csc A = 2.92\end{align*}
13. A 70 foot building casts an 100 foot shadow. What is the angle that the sun hits the building?
14. Over 2 miles (horizontal), a road rises 300 feet (vertical). What is the angle of elevation?
15. Whitney is sailing and spots a shipwreck 100 feet below the water. She jumps from the boat and swims 250 feet to reach the wreck. What is the angle of depression from the boat to the shipwreck?

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### Vocabulary Language: English

TermDefinition
inverse trigonometric function An inverse trigonometric function is a function that reverses a trigonometric function, leaving the argument of the original trigonometric function as a result.

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