SAT Math · Additional Topics in Math
Right triangle trigonometry
SOH-CAH-TOA, special right triangles, and the Pythagorean theorem.
What's tested in this subtopic
Items test the basic trig ratios (sine, cosine, tangent), the special right triangles (30-60-90 and 45-45-90), and the Pythagorean theorem. Common surface forms include finding a missing side given an angle and a side, finding an angle given two sides, or recognizing a Pythagorean triple in disguise.
Tactics that actually move your score
Memorize the special right triangle ratios. Memorize the common Pythagorean triples (3-4-5, 5-12-13, 8-15-17, 7-24-25). When you see one of these triples, you save the time of a square-root computation. The single biggest leverage point on most subtopics isn't learning more math — it's recognizing the test's preferred surface forms quickly enough that you don't burn 30 seconds re-reading the question. The first time you see a particular phrasing it might take you a full minute. The tenth time you see it, you should be reaching for your method before you've finished the sentence. Repetition is what builds that recognition. Fifteen problems in a row of the same shape is more useful than fifty mixed.
Practice questions (14)
- Easy In a right triangle, the two legs have lengths 7 and 24. What is the length of the hypotenuse?
- Easy What is the value of sin(30°)?
- Medium In a right triangle, the two legs have lengths 3 and 4. What is the length of the hypotenuse?
- Medium What is the value of sin(30°)?
- Medium In a right triangle, the two legs have lengths 3 and 4. What is the length of the hypotenuse?
- Hard What is the value of sin(45°)?
- Easy In a right triangle, the two legs have lengths 5 and 12. What is the length of the hypotenuse?
- Easy What is the value of sin(30°)?
- Medium In a right triangle, the two legs have lengths 5 and 12. What is the length of the hypotenuse?
- Medium What is the value of sin(30°)?
- Medium In a right triangle, the two legs have lengths 3 and 4. What is the length of the hypotenuse?
- Hard What is the value of sin(45°)?
- Easy In a right triangle, the two legs have lengths 5 and 12. What is the length of the hypotenuse?
- Easy What is the value of sin(90°)?
How to drill
Work through the questions above untimed. After each one, read the worked solution from start to finish — even when you got it right. Note which solution method you used, and which method we used; if they differ, ask yourself which would have been faster on test day. Speed in SAT math comes from shortening your method-selection step, not from doing arithmetic faster. Most fast students are doing the same arithmetic everyone else is — they're just spending less time deciding what to do.
Once you can clear the easy and medium items in this subtopic at 90% accuracy, attempt a timed mini-set of ten hard items at 75 seconds each. If you finish in time and score 7+ correct, you've effectively mastered the subtopic for test purposes and can move on.