SAT right triangle trigonometry rests on SOHCAHTOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. In a right triangle, the hypotenuse is always the side opposite the right angle; "opposite" and "adjacent" are relative to the acute angle you care about. These three ratios plus the cofunction identity sin(θ) = cos(90° − θ) cover every right-triangle question the Digital SAT asks in its Geometry and Trigonometry domain.
Below are two SAT-style worked examples, the method, the cofunction trap the SAT tests directly, and the Bluebook Desmos shortcut you can use on test day.
Worked example 1: right triangle with legs 6 and 8 — find tan A
SAT-style problem: In right triangle ABC, angle C is a right angle, BC = 6, and AC = 8. What is the value of tan A?
- Identify the sides relative to angle A. Opposite A is BC = 6. Adjacent to A is AC = 8. The hypotenuse (opposite the right angle at C) is AB.
- You do not need the hypotenuse for tan:
tan A = opposite/adjacent = 6/8 = 3/4. - Answer:
tan A = 3/4.
If the SAT had asked for sin A instead, you would need the hypotenuse. Pythagorean theorem: AB² = 6² + 8² = 100, so AB = 10, and sin A = 6/10 = 3/5. The 6-8-10 triangle is a scaled 3-4-5 — one of the triples the SAT reuses constantly.
The SOHCAHTOA method on the SAT, step by step
- Mark the right angle and pick your angle of interest. The acute angle the SAT names (A, B, θ) is the one "opposite" and "adjacent" refer to.
- Label the three sides. Hypotenuse = opposite the right angle. Opposite = across from your angle of interest. Adjacent = the other leg (the one that forms the angle with the hypotenuse).
- Pick the ratio the question is asking for. SOH: sin = opp/hyp. CAH: cos = adj/hyp. TOA: tan = opp/adj.
- Plug in the side lengths and simplify. The SAT prefers exact fractions over decimals — leave the answer as 3/5, not 0.6.
The cofunction identity the SAT tests: sin(θ) = cos(90° − θ)
In any right triangle, the two acute angles add to 90°, and the sine of one equals the cosine of the other. So if sin A = 3/5, then cos B = 3/5 whenever B is the other acute angle. The Digital SAT loves the abstract phrasing: In right triangle ABC, angle C is a right angle. If sin A = x, what is cos B in terms of x? Answer: cos B = x. No arithmetic needed once you spot the identity.
Worked example 2: given sin A = 3/5, find cos A
SAT-style problem: In a right triangle, the sine of acute angle A is 3/5. What is the cosine of angle A?
- Interpret sin A = 3/5 with SOHCAHTOA: the side opposite A is 3 and the hypotenuse is 5. Sketch (or picture) the right triangle with those two sides labeled.
- Find the adjacent side with the Pythagorean theorem:
adj = √(hyp² − opp²) = √(5² − 3²) = √(25 − 9) = √16 = 4. - Read cosine off the triangle:
cos A = adjacent/hypotenuse = 4/5.
This is the 3-4-5 right triangle in trig clothing — one of the triples the SAT reuses constantly. If you prefer an identity, the same answer falls out of sin²A + cos²A = 1: (3/5)² + cos²A = 1 → cos²A = 16/25 → cos A = 4/5 (positive because A is acute). Both methods land on 4/5; the SOHCAHTOA-first version stays closer to the way the rest of this page teaches the material.
Bluebook Desmos shortcut for SAT trig
Bluebook (the digital testing app) ships with Desmos, which is available on every SAT math question and supports sin, cos, and tan in degree mode. For numeric questions where you know an angle and need a side (or vice versa), type sin(30) or cos(60) after setting Desmos to degrees. That said, most SAT trig answers should come out to clean fractions — 3/5, 4/5, 5/13, etc. — so Desmos is often more useful as a sanity check than as the primary solve.
Common mistakes on SAT SOHCAHTOA questions
- Mislabeling opposite and adjacent. "Opposite" is across the triangle from your angle; "adjacent" is next to it (and not the hypotenuse). Switching them turns sin into cos and vice versa.
- Using tan when the SAT needs sin or cos. Tan does not involve the hypotenuse; sin and cos do. If the question tells you two legs and asks for sin, you have to compute the hypotenuse via Pythagorean theorem first.
- Missing the cofunction identity. When the SAT asks for cos B in terms of sin A in the same right triangle, the answer is just sin A — no computation. Trying to solve algebraically wastes 45 seconds.
- Converting to decimals unnecessarily. The SAT answer choices are exact fractions. Leave sin A as 3/5, not 0.6.
- Degree/radian mode. If you use Desmos, keep it in degree mode for SAT questions — the Geometry and Trigonometry domain uses degrees, not radians.
Not sure which SAT topics to prep first? Take our free SAT math diagnostic — a short diagnostic that tells you which Digital SAT topics need the most work, including the Geometry and Trigonometry questions built around SOHCAHTOA, right triangles, and cofunctions. No card, no email up front.
Because SOHCAHTOA questions almost always sit on top of a right triangle, our general-math walkthroughs are natural next reads: How to Find the Hypotenuse of a Right Triangle and Pythagorean Theorem Explained With Examples both cover the side-length arithmetic you need before the ratios make sense.
The three ratios are memorizable in an afternoon; the fluency is in picking the right one under time pressure and catching cofunctions without pausing. To see where SOHCAHTOA and the rest of Geometry and Trigonometry sit on your SAT readiness map right now, take the free SAT math diagnostic above. Up to 15 questions, no account required to start.