Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
The last six problems use technology and context: a Ferris wheel, tides, a ramp, a pendulum and a factoring trap.
Standards: F-TF.B.6, F-TF.B.7 · Level: Grade 12 Precalculus
Problem 1 Evaluate arcsin ★★★
Find the exact value in radians. (a) arcsin(√3/2) (b) arcsin(−1/2) (c) arcsin 0 (d) arcsin(−1)
Problem 2 Evaluate arccos ★★★
Find the exact value in radians. (a) arccos(1/2) (b) arccos(−√3/2) (c) arccos(−1) (d) arccos 0
Problem 3 Evaluate arctan ★★★
Find the exact value in radians. (a) arctan 1 (b) arctan(−√3) (c) arctan 0 (d) arctan(√3/3)
Problem 4 Domain and Range from a Graph ★★★
The graph shows y = arcsin x. State its domain and range, and give the output when x = 0.
Problem 5 Why Restrict the Domain? ★★★
Name two inputs in [0, 2π] for which sin x = 1/2, and explain why that means sine has no inverse on [0, 2π]. Then give the standard interval used for arcsin.
Problem 6 Solve Three Basic Equations ★★★
Solve on [0, 2π). (a) sin x = 1/2 (see the graph) (b) cos x = −√2/2 (c) tan x = 1
Problem 7 Compositions with Triangles ★★★
Find the exact value. (a) sin(arccos(8/17)) (b) cos(arcsin(7/25)) (c) tan(arcsin(−5/13)) (d) sin(arctan(12/5))
Problem 8 Careful with arcsin(sin x) ★★★
Evaluate. (a) arcsin(sin(5π/6)) (b) arccos(cos(−π/3)) (c) arctan(tan(3π/4)) (d) arccos(cos(7π/4))
Problem 9 Equation Solved in General ★★★
Solve 2 sin x = √3 on [0, 2π) and then write the general solution.
Problem 10 A Quadratic in Cosine ★★★
Solve 4cos2 x − 3 = 0 on [0, 2π).
Problem 11 Equation with a Multiple Angle Input ★★★
Solve sin(3x) = √3/2 on [0, 2π) and give the general solution.
Problem 12 Use a Calculator for cos x = 0.3 ★★★
Solve cos x = 0.3 on [0, 2π). Give the solutions to four decimal places. The graph shows the line y = 0.3.
Problem 13 Negative Sine Value ★★★
Solve sin x = −0.45 on [0, 2π). Give answers to four decimal places.
Problem 14 Tangent Equation with Technology ★★★
Solve 3 tan x + 5 = 0 on [0, 2π). Give answers to four decimal places.
Problem 15 Factor Instead of Dividing ★★★
Solve 2 sin x cos x = cos x on [0, 2π). Explain why dividing both sides by cos x would lose solutions.
Problem 16 Ferris Wheel at 50 Feet ★★★
A rider’s height is H(t) = 30 − 25cos(πt/20) feet, t in seconds, for one turn 0 ≤ t ≤ 40. When is the rider 50 feet above the ground? Round to the nearest tenth of a second.
Problem 17 Tide Window ★★★
Water depth at a dock is h(t) = 3.5 sin(πt/6) + 4.5 feet, where t is the hours after midnight (0 ≤ t ≤ 12). A boat needs at least 6.5 feet of water. During which times is the depth at least 6.5 feet?
Problem 18 Wheelchair Ramp Angles ★★★
A ramp 14 feet long rises 1.1 feet. (a) Find its angle of elevation using arcsin, in degrees. (b) The maximum allowed slope is 1 foot of rise per 12 feet of horizontal run. Find the corresponding angle with arctan. (c) How high can a 20-foot ramp rise at that maximum angle?
Problem 19 Swinging Pendulum ★★★
A pendulum’s angle from vertical is θ(t) = 0.35cos(5πt/4) radians, t in seconds. Find the first two times in one period when the angle is 0.2 radian, to three decimals.
Problem 20 Quadratic in Sine with an Impossible Factor ★★★
Solve 2 sin2 x + 3 sin x − 2 = 0 on [0, 2π). Explain why one factor gives no solution, then write the general solutions.
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