Precalculus Inverse Trigonometric Functions and Equations Test Solutions and Scoring (PDF)
This inverse trig test answers for Precalculus covers inverse trigonometric functions and equations (Common Core F-TF.B.6, F-TF.B.7), with a free PDF to print.
Precalculus · Common Core State Standards: F-TF.B.6, F-TF.B.7
Inverse trig test answers: worked solutions
Question 1 – Exact inverse values [2 pts]
\(\sin\left(-\frac{\pi}{6}\right)=-\frac12\). \(\cos\frac{\pi}{2}=0\).
\(-\frac{\pi}{6}\) and \(\frac{\pi}{2}\)
Scoring: 1 point each.
Question 2 – More inverse values [2 pts]
\(\tan\left(-\frac{\pi}{4}\right)=-1\). \(\sin\frac{\pi}{3}=\frac{\sqrt3}{2}\).
\(-\frac{\pi}{4}\) and \(\frac{\pi}{3}\)
Scoring: 1 point each.
Question 3 – Compositions [2 pts]
\(\sin\frac{5\pi}{6}=\frac12\) and \(\arcsin\frac12=\frac{\pi}{6}\) (not \(\frac{5\pi}{6}\)). \(\cos(\arccos0.25)=0.25\).
\(\frac{\pi}{6}\) and \(0.25\)
Scoring: 1 point each.
Question 4 – Triangle method [2 pts]
Sides 5, 12, 13 give \(\cos=\frac{12}{13}\). Sides 4, 3, 5 give \(\sin=\frac35\).
\(\frac{12}{13}\) and \(\frac35\)
Scoring: 1 point each.
Question 5 – Solve from a graph [3 pts]
(a) About \(3.7\) and \(5.8\). (b) \(\frac{\pi}{6}\). (c) Sine is negative in quadrants III and IV: \(\pi+\frac{\pi}{6}\) and \(2\pi-\frac{\pi}{6}\).
\(3.7,\ 5.8\); \(\frac{\pi}{6}\); \(x=\frac{7\pi}{6},\ \frac{11\pi}{6}\)
Scoring: 1 point for each part.
Question 6 – Solve a cosine equation [2 pts]
\(\cos x=\frac{\sqrt3}{2}\). Reference angle \(\frac{\pi}{6}\); cosine is positive in quadrants I and IV.
\(x=\frac{\pi}{6},\ \frac{11\pi}{6}\)
Scoring: 1 point for each solution.
Question 7 – General solution [2 pts]
\(\tan\frac{\pi}{3}=\sqrt3\), and tangent has period \(\pi\).
\(x=\frac{\pi}{3}+\pi k\)
Scoring: 1 point for the angle, 1 point for the period.
Question 8 – Ramp angle [3 pts]
(a) \(\theta=\arcsin\frac{3.5}{20}\approx 10.08^{\circ}\). (b) \(\arctan\frac{1}{12}\approx 4.76^{\circ}\). (c) \(10.08>4.76\).
\(10.08^{\circ}\); \(4.76^{\circ}\); yes, it is steeper
Scoring: 1 point for each part.
Question 9 – Quadratic in sine [2 pts]
\((2\sin x-1)(\sin x-1)=0\). \(\sin x=\frac12\): \(\frac{\pi}{6},\frac{5\pi}{6}\). \(\sin x=1\): \(\frac{\pi}{2}\).
\(x=\frac{\pi}{6},\ \frac{\pi}{2},\ \frac{5\pi}{6}\)
Scoring: 1 point for factoring, 1 point for all three solutions.
Total score
20 points
Key facts behind this inverse trig test answers
- 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 answers
Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.
All Precalculus chapters: Precalculus math.
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