How to Solve: ACT Math – Trigonometry (SOH CAH TOA, Unit Circle, Law of Sines/Cosines)

How to Solve: ACT Math – Trigonometry (SOH CAH TOA, Unit Circle, Law of Sines/Cosines)

Introduction

"Mastering SOH CAH TOA and the Law of Sines/Cosines can add 4-6 points to your ACT Math score—enough to turn a 25 into a 30. These questions appear on every single ACT, and they’re the easiest trig problems to solve if you know the steps. Today, you’ll learn exactly how to tackle them—no guesswork, just fast, accurate answers."

What You Need To Know First

Before diving into trigonometry, make sure you understand:
1. Right triangles – The Pythagorean theorem (a² + b² = c²) and how to label sides (opposite, adjacent, hypotenuse).
2. Basic algebra – Solving for x in equations like sin(θ) = 3/5.
3. Degrees vs. radians – The ACT uses degrees (not radians), so set your calculator to DEG mode.

Key Vocabulary

Formulas To Know

1. SOH CAH TOA (Right Triangles Only)

MEMORISE THIS – The ACT does not provide these formulas.

2. Unit Circle (Key Values)

MEMORISE THESE – The ACT expects you to know sin/cos for 0°, 30°, 45°, 60°, 90°.

3. Law of Sines (Any Triangle)

Given on exam sheet, but memorise it anyway for speed.

Formula: a / sin(A) = b / sin(B) = c / sin(C)

• a, b, c = lengths of sides opposite angles A, B, C.
• Use when you have two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

4. Law of Cosines (Any Triangle)

Given on exam sheet, but memorise it for efficiency.

Formula: c² = a² + b² – 2ab·cos(C)

• c = side opposite angle C.
• Use when you have two sides and the included angle (SAS) or all three sides (SSS).

Step-by-Step Method

For Right Triangles (SOH CAH TOA)

Step 1: Label the triangle. - Identify the hypotenuse (longest side, opposite the right angle). - Label the opposite and adjacent sides relative to the angle you’re working with.

Step 2: Choose the correct ratio. - Missing a side? Use sin, cos, or tan to set up an equation. - Missing an angle? Use sin⁻¹, cos⁻¹, or tan⁻¹ (inverse trig functions).

Step 3: Solve for the unknown. - If solving for a side: Cross-multiply and simplify. - If solving for an angle: Use the inverse function on your calculator.

Step 4: Check your answer. - Does the side length make sense? (Hypotenuse should be the longest.) - Does the angle make sense? (Should be between 0° and 90° in a right triangle.)

For Non-Right Triangles (Law of Sines/Cosines)

Step 1: Determine which law to use. - Law of Sines: Two angles + one side or two sides + a non-included angle. - Law of Cosines: Two sides + included angle or all three sides.

Step 2: Plug values into the formula. - For Law of Sines, set up a proportion (e.g., a/sin(A) = b/sin(B)). - For Law of Cosines, substitute directly (e.g., c² = a² + b² – 2ab·cos(C)).

Step 3: Solve for the unknown. - For Law of Sines, cross-multiply and solve. - For Law of Cosines, take the square root at the end.

Step 4: Verify the answer. - Does the side length fit the triangle inequality? (Sum of any two sides > third side.) - Does the angle sum to 180°? (All angles in a triangle add to 180°.)

Worked Examples

Example 1 – Basic (SOH CAH TOA)

Problem: In a right triangle, the side opposite a 30° angle is 5. What is the hypotenuse?

Step 1: Label the triangle. - Angle = 30° - Opposite side = 5 - Hypotenuse = ?

Step 2: Choose the ratio. - We have the opposite and need the hypotenuse → Use SOH (sin). - sin(30°) = Opposite / Hypotenuse

Step 3: Plug in known values. - sin(30°) = 5 / Hypotenuse - From the unit circle, sin(30°) = 1/2.

Step 4: Solve. - 1/2 = 5 / Hypotenuse - Cross-multiply: Hypotenuse = 5 × 2 = 10

Answer: The hypotenuse is 10.

What we did and why: We used SOH (sin = opposite/hypotenuse) because we had the opposite side and needed the hypotenuse. Memorizing sin(30°) = 1/2 saved time.

Example 2 – Medium (Law of Sines)

Problem: In triangle ABC, angle A = 40°, angle B = 60°, and side a = 8. What is side b?

Step 1: Find the missing angle. - Sum of angles = 180° → 40° + 60° + C = 180° → C = 80°

Step 2: Choose the law. - We have two angles and one side → Law of Sines.

Step 3: Set up the proportion. - a / sin(A) = b / sin(B) - 8 / sin(40°) = b / sin(60°)

Step 4: Solve for b. - b = (8 × sin(60°)) / sin(40°) - sin(60°) ≈ 0.866, sin(40°) ≈ 0.643 - b ≈ (8 × 0.866) / 0.643 ≈ 10.77

Answer: Side b ≈ 10.8 (rounded to one decimal place).

What we did and why: We used the Law of Sines because we had two angles and one side. Always find the missing angle first to ensure the proportion is set up correctly.

Example 3 – Exam-Style (Law of Cosines)

Problem: A triangle has sides of length 7 and 10, and the included angle is 50°. What is the length of the third side?

Step 1: Identify the given information. - Side a = 7, side b = 10, angle C = 50°. - We need side c.

Step 2: Choose the law. - We have two sides and the included angle → Law of Cosines.

Step 3: Plug into the formula. - c² = a² + b² – 2ab·cos(C) - c² = 7² + 10² – 2(7)(10)·cos(50°) - c² = 49 + 100 – 140·cos(50°)

Step 4: Calculate. - cos(50°) ≈ 0.643 - c² = 149 – 140(0.643) ≈ 149 – 89.92 ≈ 59.08 - c ≈ √59.08 ≈ 7.69

Answer: The third side is ≈ 7.7.

What we did and why: We used the Law of Cosines because we had two sides and the included angle. Always double-check the calculator mode (DEG, not RAD).

Common Mistakes

Exam Traps

1-Minute Recap

"Here’s what you need to remember for ACT Trigonometry:
1. SOH CAH TOA only works for right triangles. Label your sides: opposite, adjacent, hypotenuse.
2. Law of Sines is for two angles + one side or two sides + non-included angle. Set up a proportion.
3. Law of Cosines is for two sides + included angle or all three sides. Plug and chug.
4. Memorize the unit circle values for 0°, 30°, 45°, 60°, 90°—they come up every test.
5. Always check your calculator mode—ACT uses degrees, not radians.
6. Draw the triangle—labeling sides and angles prevents mistakes. You’ve got this. Now go crush those trig questions!