Solving Trigonometry Questions on the SAT

The method: label sides, pick the ratio

Every SAT trig question lives inside a right triangle, and every right triangle has exactly three named sides relative to whichever angle you're working with: the hypotenuse (always opposite the right angle, always the longest side), the opposite side (across from the angle you care about), and the adjacent side (next to that angle, but not the hypotenuse).

The mnemonic SOH-CAH-TOA tells you which ratio connects which pair of sides: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Use Label → Choose → Solve: first label the three sides relative to the specific angle in the problem, then choose the one ratio that involves the side you know and the side (or angle) you want, then solve algebraically.

Sine and cosine of complementary angles

In any right triangle, the two non-right angles always add up to 90° — they're complementary. That fact creates a shortcut the SAT tests directly: the side that's "opposite" one acute angle is "adjacent" to the other, so sin(θ) = cos(90° − θ) and cos(θ) = sin(90° − θ).

In practice, this means if two angles in a problem sum to 90°, the sine of one always equals the cosine of the other — no calculation required. Spotting "the two angles add to 90°" is the entire question; everything after that is just copying a number.

Special right triangles: 30-60-90 and 45-45-90

Two right triangles show up constantly because their side ratios are fixed exact values — no calculator, no decimals. In a 30-60-90 triangle, the sides opposite the 30°, 60°, and 90° angles are always in the ratio 1 : √3 : 2. So if the shortest leg (opposite 30°) is a, the longer leg (opposite 60°) is a√3, and the hypotenuse is 2a.

In a 45-45-90 triangle, both legs are equal and the ratio is 1 : 1 : √2: if each leg is a, the hypotenuse is a√2. These ratios also hand you exact trig values worth memorizing: sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2, cos 60° = 1/2, and sin 45° = cos 45° = √2/2.

Degrees, radians, and formulas to know cold

Most SAT trig questions stay in degrees, but occasionally an angle shows up in radians, or a question asks you to convert. One full rotation is 360° = 2π radians, which gives two conversion factors: multiply by π/180 to go from degrees to radians, and multiply by 180/π to go from radians to degrees.

SOH-CAH-TOA: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj.

Complementary angles: if θ + φ = 90°, then sin θ = cos φ and cos θ = sin φ.

Pythagorean identity: sin²θ + cos²θ = 1, true for any angle θ.

Special ratios: 30-60-90 sides 1 : √3 : 2 · 45-45-90 sides 1 : 1 : √2.

Run through the drills below cold — no reference sheet — until each one takes under fifteen seconds.

Common traps: which side, which angle

The most common slip is mislabeling a side — calling something "opposite" or "adjacent" without re-checking which angle it's relative to. Relabel the triangle fresh for whichever angle the question actually asks about; don't reuse labels from an earlier part of the same problem. A second trap is assuming sin(A°) = sin(B°) for the two acute angles in a right triangle — that's only true if A = B. What is always true is the complementary relationship: sin(A°) = cos(B°).

Practice questions

Here are five practice problems that help you hone your skills. Use the same method we learned earlier above to solve these problems. Remember, practice makes perfect.