Precalculus Inverse Trig Quiz: 10 Questions

Contents

Key takeaway: This precalculus inverse trig quiz contains 10 multiple-choice questions on ranges, exact values, compositions and equations. A few items show a graph or a unit circle, so you decide with a picture. The set targets standard F-TF.B.6. Every explanation calls out the range mistake that most often hides in the wrong options. Allow about 15 minutes.

Ten multiple-choice questions on inverse trigonometric functions, restricted ranges, and solving trig equations. Pick the one correct answer for each. Three questions include a figure.

PrecalculusCommon Core: F-TF.B.6 · F-TF.B.7

🎯 Inverse Trigonometric Functions and Trigonometric Equations – Precalculus math quiz

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Question 1

What is the range of \(y=\cos^{-1}x\)?

A \( \left[-\frac{\pi}{2},\frac{\pi}{2}\right] \)

B \( [0,\pi] \)

C \( [0,2\pi] \)

D \( [-\pi,\pi] \)

Question 2

What is \(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)?

A \(\frac{\pi}{3}\)

B \(\frac{5\pi}{3}\)

C \(-\frac{\pi}{3}\)

D \(-\frac{\pi}{6}\)

Question 3

What is \(\tan^{-1}(1)\)?

A \(\frac{\pi}{2}\)

B \(\frac{3\pi}{4}\)

C \(\frac{\pi}{4}\)

D \(1\)

Question 4

Which function has the graph shown?

Curve that increases and levels off at two horizontal dashed lines.

A \(y=\tan^{-1}x\)

B \(y=\sin^{-1}x\)

C \(y=\cos^{-1}x\)

D \(y=\tan x\)

Question 5

What is \(\cos^{-1}\left(\cos\frac{5\pi}{3}\right)\)?

A \(\frac{5\pi}{3}\)

B \(\frac{\pi}{3}\)

C \(-\frac{\pi}{3}\)

D \(\frac{2\pi}{3}\)

Question 6

The line \(x=\frac{1}{2}\) is drawn on the unit circle. What are the solutions of \(\cos x=\frac{1}{2}\) on \(0\le x< 2\pi\)?

A \(\frac{\pi}{3}\) and \(\frac{2\pi}{3}\)

B \(\frac{\pi}{6}\) and \(\frac{11\pi}{6}\)

C \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\)

D \(\frac{\pi}{3}\) and \(\frac{5\pi}{3}\)

Question 7

What is \(\sin\left(\cos^{-1}\frac{3}{5}\right)\)?

A \(\frac{4}{5}\)

B \(\frac{3}{5}\)

C \(\frac{3}{4}\)

D \(\frac{5}{3}\)

Question 8

The graph shows \(y=\sin x\) and the line \(y=0.8\). Which values solve \(\sin x=0.8\) on \(0\le x\le\pi\), to the nearest hundredth?

The line meets the curve at two points.

A \(0.93\) and \(4.07\)

B \(0.93\) and \(5.36\)

C \(53.13\) and \(126.87\)

D \(0.93\) and \(2.21\)

Question 9

What are all solutions of \(\tan x=1\)? Let \(k\) be any integer.

A \(x=\frac{\pi}{4}+2\pi k\)

B \(x=\frac{\pi}{4}+\pi k\)

C \(x=\pm\frac{\pi}{4}+2\pi k\)

D \(x=\frac{\pi}{4}+\frac{\pi k}{2}\)

Question 10

What is the domain of \(f(x)=\tan^{-1}x\)?

A \( [-1,1] \)

B \( \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \)

C all real numbers

D \( [0,\pi] \)

🎉 Quiz complete!

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Remember:

Sine, cosine, and tangent are not one-to-one, so each needs a restricted domain before it has an inverse.
sin⁻¹ and cos⁻¹ have domain [−1, 1]; tan⁻¹ accepts every real number.
sin⁻¹(sin x) equals x only when x is in [−π/2, π/2]; the same idea holds for the other two functions on their ranges.
A calculator gives only one solution; symmetry and the period supply the rest.
In applied problems, keep only the solutions that lie in the interval of the situation.

Which questions check ranges and values?

The range of \(\cos^{-1}x\) is \([0,\pi]\) in the first question, and the domain of \(\tan^{-1}x\) is all real numbers in the last. In between, \(\sin^{-1}(-\frac{\sqrt{3}}{2})=-\frac{\pi}{3}\) and \(\tan^{-1}(1)=\frac{\pi}{4}\). A curve that levels off at two dashed lines can only be arctan.

How do the composition questions work?

Question 5 asks for \(\cos^{-1}(\cos\frac{5\pi}{3})\). The raw angle is outside \([0,\pi]\), so the answer is \(\frac{\pi}{3}\), not \(\frac{5\pi}{3}\). Question 7 uses a 3-4-5 triangle to get \(\sin(\cos^{-1}\frac{3}{5})=\frac{4}{5}\). The temptation is to answer \(\frac{3}{5}\), which is the cosine itself.

What do the equation questions need?

For \(\cos x=\frac{1}{2}\) on a full turn, the unit circle line gives \(\frac{\pi}{3}\) and \(\frac{5\pi}{3}\). The graph with \(y=0.8\) in question 8 yields 0.93 and 2.21, which add up to \(\pi\). Finally, all solutions of \(\tan x=1\) are \(\frac{\pi}{4}+\pi k\), because tangent repeats every \(\pi\).

What is the best follow-up?

Aim for nine correct answers before the unit test. For each miss, redraw the picture the question used and say aloud what the inverse promises. Then try the quiz again after a short break.