Right Triangle Trigonometry · Sub-skill drill
SOH-CAH-TOA
In a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent — relative to one of the acute angles. The mnemonic SOH-CAH-TOA captures all three. SAT questions test this with both numeric and algebraic right triangles. The skill is identifying which side is opposite, adjacent, and hypotenuse relative to the angle of interest, then writing the matching ratio. Sketching the triangle with the angle marked makes this nearly automatic.
How this sub-skill is tested on the SAT
In a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent — relative to one of the acute angles. The mnemonic SOH-CAH-TOA captures all three. SAT questions test this with both numeric and algebraic right triangles. The skill is identifying which side is opposite, adjacent, and hypotenuse relative to the angle of interest, then writing the matching ratio. Sketching the triangle with the angle marked makes this nearly automatic.
This sub-skill sits inside the broader Right Triangle Trigonometry topic, which is part of the College Board's Additional Topics in Math content domain. Additional Topics in Math is the smallest official SAT Math domain by raw question count, but it carries outsized weight because the questions are concentrated at the harder end of each section. You will see roughly six of these per test, and they tend to separate students aiming for a 750 from students aiming for an 800. The domain covers right triangle trigonometry, circle theorems, volume formulas, complex number arithmetic, and the geometry of lines in the coordinate plane. Most of the formulas you need are listed at the start of the math section — but the test rewards students who have
Practice questions in this drill set
Below are 7 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.
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A right triangle has legs of length 12 and 16. What is the length of the hypotenuse?
- A 19
- B 28
- C 21
- D 20
Worked solution
Answer: D — 20
By the Pythagorean theorem, c^2 = a^2 + b^2 = 12^2 + 16^2 = 144 + 256 = 400. So c = √400 = 20.
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In a right triangle with legs 24 and 32 and hypotenuse 40, what is the value of sin(A) where A is the angle opposite the leg of length 24?
- A 5/3
- B 3/4
- C 4/5
- D 3/5
Worked solution
Answer: D — 3/5
SOH-CAH-TOA: sin = opposite / hypotenuse. The leg opposite angle A has length 24; the hypotenuse is 40. So sin(A) = 24/40 = 3/5.
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In a right triangle with legs 8 and 15 and hypotenuse 17, what is the value of tan(B) where B is the angle opposite the leg of length 15?
- A 15/8
- B 8/15
- C 15/17
- D 8/17
Worked solution
Answer: A — 15/8
tan = opposite / adjacent. Opposite angle B is 15; the adjacent leg is 8. So tan(B) = 15/8 = 15/8.
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A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?
- A 9
- B 11
- C 14
- D 10
Worked solution
Answer: D — 10
By the Pythagorean theorem, c^2 = a^2 + b^2 = 6^2 + 8^2 = 36 + 64 = 100. So c = √100 = 10.
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In a right triangle with legs 6 and 8 and hypotenuse 10, what is the value of sin(A) where A is the angle opposite the leg of length 6?
- A 5/3
- B 3/4
- C 4/5
- D 3/5
Worked solution
Answer: D — 3/5
SOH-CAH-TOA: sin = opposite / hypotenuse. The leg opposite angle A has length 6; the hypotenuse is 10. So sin(A) = 6/10 = 3/5.
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In a right triangle with legs 16 and 30 and hypotenuse 34, what is the value of tan(B) where B is the angle opposite the leg of length 30?
- A 15/8
- B 8/15
- C 15/17
- D 8/17
Worked solution
Answer: A — 15/8
tan = opposite / adjacent. Opposite angle B is 30; the adjacent leg is 16. So tan(B) = 30/16 = 15/8.
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A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?
- A 9
- B 11
- C 14
- D 10
Worked solution
Answer: D — 10
By the Pythagorean theorem, c^2 = a^2 + b^2 = 6^2 + 8^2 = 36 + 64 = 100. So c = √100 = 10.
Why this band assignment matters
Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.
Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.