Precalculus Inverse Trigonometric Functions and Equations Test with Point Values (PDF)

Precalculus math test: Inverse Trigonometric Functions and Equations – Maths PDF

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This inverse trig test for Precalculus covers inverse trigonometric functions and equations (Common Core F-TF.B.6, F-TF.B.7), with a free PDF to print.

This 20-point test covers inverse trig values, compositions, solving trigonometric equations and a ramp model (F-TF.B.6, F-TF.B.7). You have 45 minutes. Show your work.

Time and points

About this test

Grade: Precalculus
Chapter: Inverse Trigonometric Functions and Equations
Standards: F-TF.B.6, F-TF.B.7
Time: 45 minutes
Total: 20 points

Inverse trig test: questions

Question 1 – Exact inverse values [2 pts]

Evaluate \(\arcsin\left(-\frac12\right)\) and \(\arccos0\).

Question 2 – More inverse values [2 pts]

Evaluate \(\arctan(-1)\) and \(\arcsin\frac{\sqrt3}{2}\).

Question 3 – Compositions [2 pts]

Graph of y = arcsin x.

Evaluate \(\arcsin\left(\sin\frac{5\pi}{6}\right)\) and \(\cos(\arccos0.25)\).

Question 4 – Triangle method [2 pts]

Find \(\cos\left(\arcsin\frac{5}{13}\right)\) and \(\sin\left(\arccos\frac45\right)\).

Question 5 – Solve from a graph [3 pts]

The curve y = sin x and the line y = -1/2.

Solve \(\sin x=-\frac12\) for \(0\le x< 2\pi\). (a) Estimate the solutions from the graph. (b) State the reference angle. (c) Give the exact solutions.

Question 6 – Solve a cosine equation [2 pts]

Solve \(2\cos x-\sqrt3=0\) for \(0\le x< 2\pi\).

Question 7 – General solution [2 pts]

Write all solutions of \(\tan x=\sqrt3\).

Question 8 – Ramp angle [3 pts]

A ramp 20 feet long rising 3.5 feet.

Sofia’s ramp is \(20\) ft long and rises \(3.5\) ft. (a) Find its angle with the ground to the nearest hundredth of a degree. (b) A rule allows a slope of at most \(1\) foot of rise per \(12\) feet of run. Find that maximum angle. (c) Is Sofia’s ramp steeper than allowed?

Question 9 – Quadratic in sine [2 pts]

Solve \(2\sin^{2}x-3\sin x+1=0\) for \(0\le x< 2\pi\).

Key facts behind this inverse trig test

Key facts

arcsin and arctan have range (or closed range) centered on 0; arccos has range [0, pi].
sin(arcsin x) = x on [-1, 1], but arcsin(sin x) = x only for x in [-pi/2, pi/2].
sin^-1 x means the inverse function, not 1/sin x.
Sine and cosine equations usually have two solutions per period; tangent has one.
Factor quadratic-type trig equations before solving.

How to use this inverse trig test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All Precalculus chapters: Precalculus math.

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