Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
The last six problems apply the formulas to a treetop angle, a launched ball, interfering waves and two equations that need factoring.
Standards: F-TF.C.8, F-TF.C.9 · Level: Grade 12 Precalculus
Problem 1 Reciprocal and Quotient Values ★★★
Suppose sin θ = 5/13 and cos θ = 12/13. Find tan θ, csc θ, sec θ and cot θ.
Problem 2 Simplify with Pythagorean Identities ★★★
Simplify each expression. (a) 5 − 5sin2 x (b) 3sec2 x − 3 (c) csc2 x − cot2 x (d) cos x tan x
Problem 3 Find Sine and Tangent from Cosine ★★★
Angle θ is in Quadrant IV and cos θ = 5/13. Find sin θ and tan θ.
Problem 4 Check a Point on the Unit Circle ★★★
Show that (8/17, 15/17) lies on the unit circle, then give sin θ, cos θ, tan θ and sec θ for the angle in standard position through this point.
Problem 5 Identity or Not? ★★★
(a) Show that sin2 x + cos2 x = 1 is true for x = π/6. (b) Show that sin(x + y) = sin x + sin y is false for x = y = π/6.
Problem 6 Exact Value with a Difference Formula ★★★
Use cos(A − B) to find the exact value of cos 15°, using 15° = 45° − 30°.
Problem 7 Verify a Quotient Identity ★★★
Verify that sec x − cos x = sin x tan x.
Problem 8 Verify with Reciprocals ★★★
Verify that csc x − sin x = cos x cot x.
Problem 9 Tangent of a Sum ★★★
Find the exact value of tan(7π/12) using 7π/12 = π/3 + π/4.
Problem 10 Sum of Two Angles from Given Values ★★★
Angle A is in Quadrant I with sin A = 8/17. Angle B is in Quadrant II with cos B = −3/5. Find sin(A + B) and cos(A + B).
Problem 11 Double Angle from Cosine ★★★
Angle θ is in Quadrant II with cos θ = −7/25. Find sin 2θ and cos 2θ.
Problem 12 Double Angle from Tangent ★★★
Angle θ is in Quadrant IV with tan θ = −3/4. Find sin 2θ, cos 2θ and tan 2θ.
Problem 13 Verify a Double-Angle Quotient ★★★
Verify that 1 − cos 2xsin 2x = tan x.
Problem 14 Simplify to a Constant ★★★
Simplify sin x cos x (tan x + cot x).
Problem 15 A Longer Verification ★★★
Verify that sin x1 + cos x + 1 + cos xsin x = 2 csc x.
Problem 16 Angle of Elevation of a Treetop ★★★
From a spot on level ground, a branch is seen at angle α with tan α = 1/4. The treetop is seen at an angle α + β where tan β = 3/5. Find tan(α + β) and the angle of elevation of the treetop.
Problem 17 Range of a Launched Ball ★★★
A ball launched at v = 60 feet per second and angle θ travels a horizontal distance R = (v2/32) sin 2θ feet. If sin θ = 0.6 and θ is acute, find R. Compare with a 45° launch.
Problem 18 Interfering Sound Waves ★★★
Two speakers produce waves modeled by y1 = sin(x + π/6) and y2 = sin(x − π/6). Use the sum and difference formulas to write y1 + y2 as a single multiple of sin x.
Problem 19 Solve cos 2x = cos x ★★★
Solve cos 2x = cos x on [0, 2π). Use cos 2x = 2cos2 x − 1.
Problem 20 Solve sin 2x = sin x ★★★
Solve sin 2x = sin x on [0, 2π), and explain why dividing by sin x is not allowed.
Mis à jour le