Precalculus Inverse Trigonometric Functions and Equations Test Solutions and Scoring (PDF)

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

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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.

Worked solutions and scoring notes for the 9-question inverse trig test. Use the scoring line under each solution to award partial credit.

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]

Intersections at 7π/6 and 11π/6.

(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]

sin θ = 3.5/20.

(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

Total

20 points

Key facts behind this inverse trig test answers

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 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.

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