Inverse Trigonometric Functions Test – Precalculus Math (Printable PDF)

Question 1 Exact arcsin and arccos (2 points)

(a) arcsin(−√2/2) (1 point) (b) arccos(−1/2) (1 point)

Question 2 arctan and Ranges (2 points)

(a) arctan √3 (1 point) (b) State the domain and the range of y = arctan x. (1 point)

Question 3 Compositions (2 points)

(a) cos(arcsin(8/17)) (1 point) (b) arcsin(sin(4π/3)) (1 point)

Question 4 Read Solutions from a Graph (2 points)

The graph shows y = cos x and the line y = −1/2 on [0, 2π].

1. How many solutions does cos x = −1/2 have on [0, 2π]? (1 point)
2. Find them exactly. (1 point)

Question 5 A Quadratic in Sine (3 points)

Solve 2 sin² x + 3 sin x + 1 = 0 on [0, 2π).

Question 6 Tangent of a Double Angle (3 points)

Solve tan(2x) = √3 on [0, 2π), then write the general solution.

Question 7 Calculator Solution (2 points)

Solve 4cos x = 1.8 on [0, 2π). Round to four decimal places.

Question 8 Wave Buoy Height (3 points)

A buoy’s height above the sea floor is h(t) = 6 + 2.4cos(πt/5) feet, t in seconds, for one period 0 ≤ t ≤ 10.

1. Write the equation to solve for the times when the height is 8 feet. (1 point)
2. Find the first time, to two decimals. (1 point)
3. Find the second time in the period. (1 point)

Question 9 A False Identity (1 point)

Explain why arcsin(sin x) = x is false for x = π.