ACT Math: SOH-CAH-TOA and Basic Trigonometric Ratios

SOH-CAH-TOA breaks down as: SOH (Sine=Opposite/Hypotenuse), CAH (Cosine=Adjacent/Hypotenuse), TOA (Tangent=Opposite/Adjacent). In a right triangle, identify which angle you're working with, then label the opposite side (across from your angle), the adjacent side (next to your angle), and the hypotenuse (longest side). Choose the ratio that uses the sides you know and the side you're solving for. Memorize SOH-CAH-TOA and you can solve any right triangle trig problem on the ACT.

Example: A right triangle has angle 30° with opposite side 5 and hypotenuse unknown. Use SOH (Sine=Opposite/Hypotenuse): sin(30°)=5/h. Since sin(30°)=0.5, you get 0.5=5/h, so h=10. The hypotenuse is 10.

Mistake 1: Identifying the wrong side as opposite or adjacent. The opposite side is always across from your target angle; the adjacent side is always next to it (but not the hypotenuse). Draw a small diagram and label clearly. Mistake 2: Using the wrong trigonometric function. If you know opposite and hypotenuse, use sine, not tangent. Mistake 3: Forgetting that sin and cos outputs are decimals, not angles. sin(30°)=0.5, not 0.5°. Double-check your angle and side labels before you commit to a trig function.

Most errors happen because students rush through the setup. Spend 15 seconds drawing and labeling, then choose your function. This prevents mistakes.

For each problem, (1) identify the angle, (2) label opposite, adjacent, and hypotenuse, (3) choose SOH/CAH/TOA, (4) set up the equation, (5) solve. Problem 1: Angle 40°, opposite=7, find hypotenuse. Problem 2: Angle 60°, hypotenuse=10, find opposite. Problem 3: Angle 45°, adjacent=5, find opposite. Problem 4: Find angle if opposite=6 and hypotenuse=12. Problem 5: Angle 35°, opposite=8, find adjacent. Work through all five, showing every step, and check your answers against a calculator.

If you miss any problem, redo it and identify which step—setup or calculation—broke. Drill until all five are correct.

Right triangle trigonometry appears on most ACT Math tests. These problems are straightforward once you know SOH-CAH-TOA. Mastering this one memory device and the three trig functions gives you 1-2 free points per test.

Spend 30 minutes today learning SOH-CAH-TOA and drilling the five problems. By test day, you'll recognize when to use trig and solve these problems in seconds.