Inverse Trigonometric Functions and Trigonometric Equations: Precalculus Math Quiz – PDF Download
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\( \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \)
Call 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.
