ACT Math · Worked Examples
ACT Math Trigonometry: 5 Practice Problems Solved Step by Step
Trigonometry accounts for roughly 7–10% of the ACT Math section — about 4 to 6 questions per test. That's enough to meaningfully move your score. And unlike geometry, trig questions often feel harder than they actually are once you know what the ACT is really testing.
Below are 5 ACT-style trig problems, worked step by step. Each one targets a different skill the ACT actually tests. Use these as a warm-up before jumping into the full practice quizzes.
Trig Quick Reference
sin θopposite / hypotenuse
cos θadjacent / hypotenuse
tan θopposite / adjacent
sin²θ + cos²θ= 1
sin(90° − θ)= cos θ (co-function)
Radians ↔ Degreesπ rad = 180°
Problem 1 — SOHCAHTOA (Fundamental)
Problem 1
In a right triangle, the side opposite angle A has length 5 and the hypotenuse has length 13. What is the value of cos A?
1
Identify what you know: opposite = 5, hypotenuse = 13.
2
Find the adjacent side using the Pythagorean theorem: adjacent² = 13² − 5² = 169 − 25 = 144 → adjacent = 12.
3
Apply the definition: cos A = adjacent / hypotenuse = 12/13.
Answer: cos A = 12/13
Practice · Fundamental
ACT Trigonometry — Quiz 1
6 SOHCAHTOA and basic trig ratio questions. Start here if you're new to ACT trig.
Start Quiz 1 →
Problem 2 — Finding an Angle (Fundamental → Intermediate)
Problem 2
In a right triangle, the side adjacent to angle B is 8 and the hypotenuse is 10. Which expression gives the measure of angle B?
1
You have adjacent and hypotenuse — that's the cosine ratio: cos B = 8/10 = 0.8.
2
To find the angle, use the inverse: B = cos⁻¹(0.8).
3
The ACT almost always asks for the expression, not the computed value. The correct answer is B = cos⁻¹(4/5).
Answer: B = cos⁻¹(4/5)
Problem 3 — Co-Function Identity (Intermediate)
Problem 3
If cos(40°) = 0.766, what is sin(50°)?
1
Recognize that 40° + 50° = 90°. These are complementary angles.
2
Apply the co-function identity: sin(90° − θ) = cos(θ). So sin(50°) = sin(90° − 40°) = cos(40°).
3
Therefore sin(50°) = 0.766.
Answer: sin(50°) = 0.766
Practice · Intermediate
ACT Trigonometry — Quizzes 3 & 4
Identities, unit circle values, and radians. 7 and 6 questions each.
Start Quiz 3 →
Problem 4 — Unit Circle (Intermediate → Advanced)
Problem 4
What is the exact value of tan(3π/4)?
1
Convert to degrees if easier: 3π/4 × (180°/π) = 135°.
2
135° is in Quadrant II. Its reference angle is 180° − 135° = 45°.
3
tan(45°) = 1. In Quadrant II, tangent is negative (sin is positive, cos is negative, so sin/cos is negative).
Answer: tan(3π/4) = −1
Problem 5 — Trig in Word Context (Advanced)
Problem 5
A ladder 20 feet long leans against a vertical wall. The base of the ladder is 8 feet from the wall. What is the sine of the angle the ladder makes with the ground, to the nearest hundredth?
1
Draw the triangle: hypotenuse = 20 (ladder), adjacent = 8 (ground distance), angle θ is at the base.
2
Find the opposite side (wall height): opposite² = 20² − 8² = 400 − 64 = 336 → opposite = √336 ≈ 18.33 ft.
3
sin θ = opposite / hypotenuse = 18.33 / 20 ≈ 0.917.
Answer: sin θ ≈ 0.92
All 6 ACT Trigonometry Quizzes
Intermediate
Quiz 3
Co-function identities, Pythagorean identity, special angles. 7 questions.
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