Precalculus Inverse Trigonometric Functions and Equations Worksheet (PDF)

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

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

These 20 problems practice inverse trigonometric functions and equations (Common Core F-TF.B.6 and F-TF.B.7). You will evaluate arcsin, arccos and arctan, compose trig functions, and solve equations on an interval and in general.

Precalculus · Common Core State Standards: F-TF.B.6, F-TF.B.7

Inverse trig worksheet: exercises 1 to 10

Exercise 1 – Exact values of inverse trig

Evaluate \(\arcsin\frac12\), \(\arccos\frac{\sqrt3}{2}\) and \(\arctan1\).

Exercise 2 – Negative inputs

Evaluate \(\arcsin\left(-\frac{\sqrt2}{2}\right)\), \(\arccos\left(-\frac12\right)\) and \(\arctan(-\sqrt3)\).

Exercise 3 – Domain and range

Graph of y = arcsin x.

State the domain and range of \(y=\arcsin x\) using the graph.

Exercise 4 – Calculator values

Use a calculator in radian mode to find \(\arcsin0.4\) and \(\arctan2.5\) to four decimal places.

Exercise 5 – Inverse properties

Evaluate \(\sin(\arcsin0.3)\), \(\cos(\arccos(-0.8))\) and \(\arcsin\left(\sin\frac{\pi}{6}\right)\).

Exercise 6 – Outside the principal range

Evaluate \(\arcsin\left(\sin\frac{7\pi}{6}\right)\) and \(\arccos\left(\cos\left(-\frac{\pi}{3}\right)\right)\).

Exercise 7 – Cosine of an arcsine

Find \(\cos\left(\arcsin\frac35\right)\).

Exercise 8 – Tangent of an arccosine

Find \(\tan\left(\arccos\frac{5}{13}\right)\).

Exercise 9 – Sine of an arctangent

Find \(\sin\left(\arctan\frac34\right)\).

Exercise 10 – Solve sin x = 1/2

Two points have y-coordinate 1/2.

Solve \(\sin x=\frac12\) for \(0\le x< 2\pi\).

Exercises 11 and up

Exercise 11 – Solve cos x

Solve \(\cos x=-\frac{\sqrt2}{2}\) for \(0\le x< 2\pi\).

Exercise 12 – Solve tan x

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

Exercise 13 – General solutions

Write all solutions of \(\sin x=\frac{\sqrt3}{2}\).

Exercise 14 – Rearrange first

Solve \(2\sin x-\sqrt2=0\) for \(0\le x< 2\pi\).

Exercise 15 – Quadratic in sine

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

Exercise 16 – A non-special value

Graph of cos x and y = 0.3.

Solve \(\cos x=0.3\) for \(0\le x< 2\pi\) to three decimal places.

Exercise 17 – Double angle inside

Solve \(\sin2x=\frac{\sqrt3}{2}\) for \(0\le x< 2\pi\).

Exercise 18 – Ramp angle

A ramp 12 feet long rises 2 feet.

A loading ramp of length \(12\) ft rises \(2\) ft. Find the angle it makes with the ground, to the nearest tenth of a degree.

Exercise 19 – Ladder

A 15-foot ladder with its base 4 feet from the wall.

A \(15\)-ft ladder has its base \(4\) ft from a wall. Find the angle between the ladder and the ground to the nearest tenth of a degree.

Exercise 20 – Ferris wheel timing

A rider’s height is \(h(t)=30-25\cos\left(\frac{\pi}{15}t\right)\) ft. Find the first two times in \([0,30]\) when the height is \(40\) ft, to the nearest hundredth of a second.

Key facts behind this inverse trig worksheet

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 worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

All Precalculus chapters: Precalculus math.

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