# 3.5.1: Exercises 3.5

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## Terms and Concepts

Exercise $$\PageIndex{1}$$

Explain why the range of $$\arcsin{(x)}$$ is restricted to $$[-\frac{\pi}{2},\frac{\pi}{2}]$$.

The range of $$\arcsin{(x)}$$ is restricted so that every input only has one output.

Exercise $$\PageIndex{2}$$

Explain how the unit circle helps evaluate inverse trigonometric functions.

You can look through the coordinates to see when the regular function is equal to the input of the inverse function.

Exercise $$\PageIndex{3}$$

True or false: $$\sin^2{(5x)} + \cos^2{(5x)} =5$$. If false, correct the statement.

False, $$\sin^2{(5x)} + \cos^2{(5x)} = 1$$.

## Problems

Evaluate each statement given in exercises $$\PageIndex{4}$$ - $$\PageIndex{8}$$.

Exercise $$\PageIndex{4}$$

$$\arcsin{(\frac{\sqrt{3}}{2})}$$

$$\frac{\pi}{3}$$

Exercise $$\PageIndex{5}$$

$$\arccos{(\frac{-\sqrt{3}}{2})}$$

$$\frac{5\pi}{6}$$

Exercise $$\PageIndex{6}$$

$$\arctan{(-1)}$$

$$\frac{-\pi}{4}$$

Exercise $$\PageIndex{7}$$

$$\arctan{(\sqrt{3})}$$

$$\frac{\pi}{3}$$

Exercise $$\PageIndex{8}$$

$$\arcsin{(0)}$$

$$0$$

Use your knowledge of the trigonometric functions and their relationships to right triangles to answer the questions in exercises $$\PageIndex{9}$$ – $$\PageIndex{18}$$.

Exercise $$\PageIndex{9}$$

Consider a right triangle with a hypotenuse of length 5 inches. If one of the sides measures 3 inches, what is the tangent of the angle that is opposite of that side?

$$\pm\frac{3}{4}$$

Exercise $$\PageIndex{10}$$

Consider a right triangle with a hypotenuse of length 5 inches. If one of the sides measures 2 inches, what is the sine of the angle that is opposite that side?

$$\frac{2}{5}$$

Exercise $$\PageIndex{11}$$

Imagine a circle with a radius of 2 units centered around the origin. What are the angles associated with the intersection(s) of this circle and the line $$x=1$$?

$$\theta = \frac{\pi}{3}, \frac{5\pi}{3}$$

Exercise $$\PageIndex{12}$$

Sketch a right triangle that can be associated with $$\theta = -\frac{2\pi}{3}$$ and evaluate $$\tan{(\theta)}$$.

$$\tan{(-\frac{2\pi}{3})} = \sqrt{3}$$

Exercise $$\PageIndex{13}$$

Sketch a right triangle that can be associated with $$\theta = \frac{5\pi}{4}$$ and evaluate $$\cot{(\theta)}$$.

$$\cot{(\frac{5\pi}{4})} =1$$

Exercise $$\PageIndex{14}$$

Sketch a right triangle that can be associated with $$\theta = -\frac{\pi}{6}$$ and evaluate $$\sin{(\theta)}$$.

$$\sin{(-\frac{\pi}{6})} =-\frac{1}{2}$$

Exercise $$\PageIndex{15}$$

Suppose that $$\sin{(\theta)}=\frac{3}{5}$$. What are all possible values of $$\cos{(\theta)}$$?

$$\frac{4}{5}$$ and $$-\frac{4}{5}$$.

Exercise $$\PageIndex{16}$$

Suppose that $$\cos{(\theta)}=\frac{5}{13}$$. What are all possible values of $$\tan{(\theta)}$$?

$$\frac{12}{5}$$ and $$-\frac{12}{5}$$.

Exercise $$\PageIndex{17}$$

Suppose that $$\tan{(\theta)}=-1$$. What are all possible values of $$\csc{(\theta)}$$?

$$\sqrt{2}$$ and $$-\sqrt{2}$$

Exercise $$\PageIndex{18}$$

A classic calculus problem involves a ladder leaning against a wall. The base of the ladder starts sliding away from the wall causing the top of the ladder to slide down the wall. If you know that the ladder has a length of 13 feet, find the value of $$\cos{(\theta)}$$ when the base of the ladder is 12 feet away from the wall.

$$\frac{12}{13}$$ if $$\theta$$ is the angle between the ladder and the floor; $$\frac{5}{13}$$ if $$\theta$$ is the angle between the ladder and the wall