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 \([-\frac{\pi}{2},\frac{\pi}{2}]\).

Answer: 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.

Answer: 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.

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

Problems

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

Exercise \(\PageIndex{4}\)

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

Answer: \(\frac{\pi}{3}\)

Exercise \(\PageIndex{5}\)

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

Answer: \(\frac{5\pi}{6}\)

Exercise \(\PageIndex{6}\)

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

Answer: \(\frac{-\pi}{4}\)

Exercise \(\PageIndex{7}\)

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

Answer: \(\frac{\pi}{3}\)

Exercise \(\PageIndex{8}\)

\(\arcsin{(0)}\)

Answer: \(0\)

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?

Answer: \(\pm\frac{3}{4}\)

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?

Answer: \(\frac{2}{5}\)

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 \(x=1\)?

Answer: \(\theta = \frac{\pi}{3}, \frac{5\pi}{3}\)

Exercise \(\PageIndex{12}\)

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

Answer

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

Exercise \(\PageIndex{13}\)

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

Answer

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

Exercise \(\PageIndex{14}\)

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

Answer

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

Exercise \(\PageIndex{15}\)

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

Answer: \(\frac{4}{5}\) and \(-\frac{4}{5}\).

Exercise \(\PageIndex{16}\)

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

Answer: \(\frac{12}{5}\) and \(-\frac{12}{5}\).

Exercise \(\PageIndex{17}\)

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

Answer: \(\sqrt{2}\) 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.

Answer: \(\frac{12}{13}\) 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