Key takeaway: This precalculus inverse trig worksheet provides 24 printable problems on arcsin, arccos, arctan and trigonometric equations. Problems 1 to 9 evaluate inverse trig values and compositions, problems 10 to 18 solve equations on one turn and in general form, and problems 19 to 24 use ramps, ladders, daylight and voltage. Standard F-TF.B.7 applies to the models. The answer key closes the PDF, with radian mode needed on the calculator.
These problems practice evaluating arcsin, arccos, and arctan, understanding restricted domains, and solving trigonometric equations on an interval and in general (F-TF.B.6 and F-TF.B.7).
Early problems use exact values; later ones use a calculator in radian or degree mode. The last six problems place trig equations in real situations such as ramps, ladders, daylight, and household voltage.
Precalculus Inverse Trig Worksheet: problems 1–5
1. Evaluate arcsin
Find the exact value of each expression.
- \(\sin^{-1}(\frac{1}{2})\)
- \(\sin^{-1}(-\frac{1}{2})\)
- \(\sin^{-1}(\frac{\sqrt{3}}{2})\)
- \(\sin^{-1}(-1)\)
2. Evaluate arccos
Find the exact value of each expression.
- \(\cos^{-1}(\frac{\sqrt{2}}{2})\)
- \(\cos^{-1}(-\frac{1}{2})\)
- \(\cos^{-1}(0)\)
- \(\cos^{-1}(-1)\)
3. Evaluate arctan
Find the exact value of each expression.
- \(\tan^{-1}(1)\)
- \(\tan^{-1}(-1)\)
- \(\tan^{-1}(\sqrt{3})\)
- \(\tan^{-1}(-\frac{1}{\sqrt{3}})\)
4. Graph of arctan
The graph shows \(y=\tan^{-1}x\). State its domain, its range, and the equations of its two horizontal asymptotes. Then find \(\tan^{-1}(0)\).
5. Restricted cosine
The graph of \(y=\cos x\) is shown with the thick piece used to define \(\cos^{-1}x\). State the interval of the thick piece, then find each value.
- \(\cos^{-1}(\cos\frac{\pi}{3})\)
- \(\cos^{-1}(\cos(-\frac{\pi}{3}))\)
- \(\cos^{-1}(\cos\frac{4\pi}{3})\)
Precalculus Inverse Trig Worksheet: problems 6–10
6. Compositions that cancel
Simplify each expression, or write undefined.
- \(\sin(\sin^{-1}(0.3))\)
- \(\cos(\cos^{-1}(-0.8))\)
- \(\tan(\tan^{-1}(25))\)
- \(\sin(\sin^{-1}(1.2))\)
7. Arcsine of a sine
Find each value.
- \(\sin^{-1}(\sin\frac{\pi}{4})\)
- \(\sin^{-1}(\sin\frac{3\pi}{4})\)
- \(\sin^{-1}(\sin\frac{7\pi}{6})\)
- \(\sin^{-1}(\sin(-\frac{2\pi}{3}))\)
8. Triangle compositions
The triangle shows an angle \(\theta\) with \(\cos\theta=\frac{3}{5}\). Find the exact value of each expression.
- \(\sin(\cos^{-1}\frac{3}{5})\)
- \(\cos(\sin^{-1}\frac{5}{13})\)
- \(\tan(\cos^{-1}\frac{4}{5})\)
- \(\tan(\sin^{-1}(-\frac{3}{5}))\)
9. Calculator practice
With the calculator in radian mode, evaluate each expression to the nearest hundredth.
- \(\sin^{-1}(0.35)\)
- \(\cos^{-1}(0.2)\)
- \(\tan^{-1}(4)\)
- \(\cos^{-1}(-0.65)\)
10. Solve a sine equation
Solve \(\sin x=\frac{\sqrt{3}}{2}\) for \(0\le x< 2\pi\).
Precalculus Inverse Trig Worksheet: problems 11–15
11. Solve a cosine equation
Solve \(\cos x=-\frac{\sqrt{2}}{2}\) for \(0\le x< 2\pi\).
12. Solve a tangent equation
Solve \(\tan x=-1\) for \(0\le x< 2\pi\).
13. Equation with a unit circle
Solve \(2\cos x+1=0\) for \(0\le x< 2\pi\). The vertical line on the unit circle shows where \(\cos x\) has the needed value.
14. Sine equation by calculator
Solve \(3\sin x=1.2\) for \(0\le x< 2\pi\). Round to the nearest hundredth.
15. Cosine equation by calculator
Solve \(\cos x=0.3\) for \(0\le x< 2\pi\). Round to the nearest hundredth.
Precalculus Inverse Trig Worksheet: problems 16–20
16. General solutions
Write all real solutions. Let \(k\) be any integer.
- \(\sin x=\frac{1}{2}\)
- \(\cos x=0\)
- \(\tan x=\sqrt{3}\)
17. Quadratic in sine
Solve \(2\sin^{2}x-\sin x-1=0\) for \(0\le x< 2\pi\).
18. Multiple-angle equation
Solve \(\sin 2x=\frac{\sqrt{2}}{2}\) for \(0\le x< 2\pi\).
19. Wheelchair ramp
A wheelchair ramp is 12 ft long and rises 1 ft. Find the angle \(\theta\) the ramp makes with the ground, in degrees to the nearest hundredth.
20. Ladder angle
The base of a 16-ft ladder is 4 ft from a wall.
- Find the angle the ladder makes with the ground, to the nearest tenth of a degree.
- How far from the wall should the base be to make a \(70^{\circ}\) angle? Round to the nearest tenth of a foot.
Precalculus Inverse Trig Worksheet: problems 21–24
21. Daylight hours
The hours of daylight in a northern city are \(D(t)=12+4\sin(\frac{2\pi}{365}(t-80))\), where \(t\) is the day of the year. Find the days when \(D=14\) and describe when the city has more than 14 hours of daylight.
22. Spring height
A weight on a spring has height \(h(t)=20+6\cos(\frac{2\pi t}{3})\) inches above the floor, with \(t\) in seconds. Find all times in \(0\le t< 3\) when the weight is 23 in high, and when it is 16 in high (nearest hundredth).
23. Household voltage
The voltage of a US wall outlet is \(V(t)=170\sin(120\pi t)\) volts, with \(t\) in seconds. Find the first two times in one cycle when \(V=100\) (in milliseconds, nearest hundredth). Then state the length of one cycle.
24. Sun angle
A 40-ft flagpole casts a 55-ft shadow.
- Find the angle of elevation of the sun to the nearest tenth of a degree.
- How long is the shadow when the sun is \(60^{\circ}\) above the horizon? Round to the nearest tenth of a foot.
Which problems practice exact values?
Problems 1 to 3 ask for exact values such as \(\sin^{-1}(\frac{1}{2})\), \(\cos^{-1}(\frac{\sqrt{2}}{2})\) and \(\tan^{-1}(1)\). Problem 4 reads domain, range and two horizontal asymptotes from the graph of arctan. Then problem 5 highlights the thick piece of cosine that defines its inverse. Keep the ranges in mind before you answer.
How do the composition problems trap students?
Problem 6 includes undefined cases, since \(\sin(\sin^{-1}(1.3))\) has no value. Problem 7 compares \(\sin^{-1}(\sin\frac{\pi}{4})\) with \(\sin^{-1}(\sin\frac{3\pi}{4})\), and the two answers differ. Meanwhile, problem 8 gives a triangle with \(\cos\theta=\frac{3}{5}\) and asks for exact mixed values.
What do the equation problems require?
Problems 10 to 13 use exact angles, for example \(\sin x=\frac{\sqrt{3}}{2}\). Problems 14 and 15 switch to the calculator for \(3\sin x=1.2\) and \(\cos x=0.3\), where a second angle follows from symmetry. Problem 17 factors \(2\sin^{2}x-\sin x-1=0\), and problem 18 halves the angle after solving \(\sin 2x=\frac{\sqrt{2}}{2}\).
Where do the applications appear?
A ramp 12 feet long rises 1 foot in problem 19. A 16-foot ladder in problem 20 stands 4 feet from a wall. Finally, problem 24 finds the sun’s angle from a 40-foot flagpole and a 55-foot shadow.
How should you check your solutions?
Substitute each equation answer into the original, and confirm that you found every solution on the interval. Count the angles: sine and cosine usually give two, and tangent gives two over a full turn. Then review the key and move to the quiz.
Printable PDF · Worksheet
Precalculus Inverse Trigonometric Functions and Trigonometric Equations worksheet
Download the PDF
See every chapter on the Precalculus math page.
