SAT Trigonometry Basics: Sine, Cosine, and Tangent on Right Triangles

SOHCAHTOA is a mnemonic for the three main trigonometric ratios in a right triangle. SOH: Sine=Opposite/Hypotenuse. CAH: Cosine=Adjacent/Hypotenuse. TOA: Tangent=Opposite/Adjacent. For a right triangle with angle θ, the opposite side is the side across from θ, the adjacent side is the side next to θ (not the hypotenuse), and the hypotenuse is the longest side opposite the right angle. For example, in a right triangle with angle 30°, if the opposite side is 1 and the hypotenuse is 2, then sin(30°)=1/2=0.5. These ratios are constant for a given angle, regardless of the triangle's size. A memorization routine: Write SOHCAHTOA on a flashcard. On the back, draw a right triangle labeled opposite, adjacent, and hypotenuse for angle θ. Spend 3 minutes daily for one week reviewing this card until SOHCAHTOA is automatic. By test day, you should be able to recall these ratios instantly without the mnemonic.

Special angle values worth memorizing: sin(30°)=1/2, cos(30°)=sqrt(3)/2, tan(30°)=1/sqrt(3). sin(45°)=1/sqrt(2), cos(45°)=1/sqrt(2), tan(45°)=1. sin(60°)=sqrt(3)/2, cos(60°)=1/2, tan(60°)=sqrt(3). These appear frequently on the SAT, and knowing them saves time compared to calculating each time.

To find an unknown side in a right triangle when you know one side and an angle: (1) Identify which sides are opposite, adjacent, and hypotenuse relative to the known angle. (2) Determine which ratio(s) involve the known side and the unknown side. (3) Set up the equation and solve. Example: In a right triangle with a 30° angle and hypotenuse 10, find the opposite side. sin(30°)=opposite/10, so 1/2=opposite/10, so opposite=5. Another example: With a 45° angle and adjacent side 8, find the opposite side. tan(45°)=opposite/8, so 1=opposite/8, so opposite=8. A systematic process for right triangle problems: (1) Draw the triangle and label all known sides and angles. (2) Identify the unknown you are solving for. (3) Determine which trig ratio(s) connect the known and unknown. (4) Write the equation. (5) Solve for the unknown. Writing out all steps prevents errors and makes verification easier.

Three micro-examples: (1) Right triangle, angle 60°, hypotenuse 10, find opposite side. sin(60°)=opposite/10 gives sqrt(3)/2=opposite/10, so opposite=5sqrt(3). (2) Angle 45°, adjacent side 6, find opposite. tan(45°)=opposite/6 gives 1=opposite/6, so opposite=6. (3) Opposite side 5, hypotenuse 13, find the angle. sin(θ)=5/13 gives θ=arcsin(5/13)≈23.1°.

Inverse trigonometric functions (arcsin, arccos, arctan) find an angle when you know a ratio. If sin(θ)=0.5, then θ=arcsin(0.5)=30°. On most SAT calculators, these functions are labeled sin^(-1), cos^(-1), tan^(-1). If you know opposite=5 and hypotenuse=13, sin(θ)=5/13, so θ=arcsin(5/13) using your calculator. Ensure your calculator is in degree mode (not radian mode) when solving SAT problems unless specifically told to use radians. A common error: forgetting to check the calculator mode and getting an answer in radians that does not match the expected degree answer. Before using inverse trig functions, check that your calculator is in the correct mode (usually degrees for SAT). If you get an unexpectedly small or large angle, verify the mode first.

Inverse functions are particularly useful when two sides are known and the angle is unknown. This setup appears frequently on SAT problems involving angles in real-world contexts (ladder leaning against a wall, angle of elevation/depression, etc.).

A 1-week trig drill solidifies these skills. Days 1-2: Memorize SOHCAHTOA and special angle values. Days 3-4: Solve right triangles finding missing sides using trig ratios. Days 5-6: Use inverse trig functions to find angles when sides are known. Day 7: Mix all types and identify errors. Common errors: (1) Confusing opposite and adjacent (carefully label your triangle); (2) Using the wrong ratio (double-check your SOHCAHTOA application); (3) Calculator in wrong mode (check radians vs. degrees); (4) Forgetting to rationalize or simplify radical answers; (5) Misidentifying which angle is which (label carefully). Track which errors you make most and drill those specifically.

On test day, when you encounter a right triangle problem, draw it out and label all known sides and angles. Identify the unknown. Determine the relevant trig ratio. Set up and solve. Verify your answer makes intuitive sense (does the angle seem reasonable? Is the side length proportional to other sides?). This careful approach prevents the majority of trig errors.