Key takeaway: This precalculus inverse trig test is a printable PDF with nine questions on exact inverse values, compositions and trigonometric equations. It rewards students who respect the range of each inverse and who find every solution on an interval. The equations come from standard F-TF.B.7. Use radian mode. Show the steps that lead to each angle. Plan on about 45 minutes.
This test covers inverse trigonometric functions, restricted domains, and trigonometric equations. It has 9 questions worth 20 points and should take about 45 minutes. A calculator is allowed. Give exact answers in terms of \(\pi\) unless a rounded answer is requested.
- Grade: Precalculus
- Chapter: Inverse Trigonometric Functions and Trigonometric Equations
- Common Core: F-TF.B.6, F-TF.B.7
- Time: 45 minutes
- Total: 20 points (Question 1: 3 pts; Question 2: 2 pts; Question 3: 2 pts; Question 4: 2 pts; Question 5: 2 pts; Question 6: 2 pts; Question 7: 2 pts; Question 8: 3 pts; Question 9: 2 pts)
Precalculus Inverse Trig Test: questions 1–5
1. Exact inverse values (3 points)
- Evaluate \(\sin^{-1}(-\frac{\sqrt{2}}{2})\). (1 point)
- Evaluate \(\cos^{-1}(-\frac{\sqrt{3}}{2})\). (1 point)
- Evaluate \(\tan^{-1}(-\sqrt{3})\). (1 point)
2. Compositions (2 points)
- Find \(\cos^{-1}(\cos\frac{7\pi}{4})\). (1 point)
- Find \(\sin^{-1}(\sin\frac{4\pi}{3})\). (1 point)
3. Restricted sine (2 points)
The graph shows \(y=\sin x\).
- State the interval on which sine is restricted to define \(\sin^{-1}x\). (1 point)
- Explain why sine has no inverse function on all real numbers. (1 point)
4. Triangle compositions (2 points)
- Find the exact value of \(\sin(\cos^{-1}\frac{5}{13})\). (1 point)
- Find the exact value of \(\cos(\sin^{-1}(-\frac{4}{5}))\). (1 point)
5. Exact equation (2 points)
Solve \(2\cos x-\sqrt{3}=0\) for \(0\le x< 2\pi\).
Precalculus Inverse Trig Test: questions 6–9
6. Sine on the unit circle (2 points)
The unit circle shows the line \(y=-\frac{1}{2}\).
- Find all solutions of \(\sin x=-\frac{1}{2}\) for \(0\le x< 2\pi\). (1 point)
- Write all real solutions. (1 point)
7. Equation by calculator (2 points)
Solve \(5\cos x=2\) for \(0\le x< 2\pi\). Round each answer to the nearest hundredth.
8. Dock depth (3 points)
The depth of water at a dock is \(d(t)=8+3\sin(\frac{\pi t}{6})\) feet, where \(t\) is the number of hours after 6 a.m., for \(0\le t\le 12\).
- Find all times when the depth is 9.5 ft. (2 points)
- Describe when the depth is at least 9.5 ft. (1 point)
9. Quadratic equation (2 points)
Solve \(2\sin^{2}x+\sin x-1=0\) for \(0\le x< 2\pi\).
What do the first questions test?
You begin with exact values such as \(\sin^{-1}(-\frac{\sqrt{2}}{2})\) and a cosine inverse. Next come compositions like \(\cos^{-1}(\cos\frac{7\pi}{4})\) and \(\sin^{-1}(\sin\frac{4\pi}{3})\). Both angles fall outside the restricted piece, so convert them before you answer. A graph of sine with a thick middle piece then asks for the interval used.
Which questions use triangles and the unit circle?
Item 4 finds \(\sin(\cos^{-1}\frac{5}{13})\), so place 5 and 12 as the legs of a triangle. Item 5 solves \(2\cos x-\sqrt{3}=0\), and item 6 reads solutions off the line \(y=-\frac{1}{2}\) on a unit circle. Remember that each horizontal line cuts the circle twice.
How do the last questions differ?
Item 7 requires a calculator: \(5\cos x=2\) has two answers, found by the inverse and by symmetry. The dock depth \(d(t)=8+3\sin(\frac{\pi t}{6})\) in item 8 asks when the water reaches a given level. Finally, item 9 factors a quadratic in sine.
Check each solution against the interval before you move on.
How can you prepare for this paper?
Memorize the three ranges, then test yourself with five tricky compositions. Meanwhile, practice drawing a quick unit circle for each equation. A tiny sketch tells you how many answers to expect.
Printable PDF · Test
Precalculus Inverse Trigonometric Functions and Trigonometric Equations test
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