Use right triangle trigonometry (sin, cos, tan) to solve for sides/angles. | Right triangles and trigonometry | SAT Math

Match the angle first, then choose sine, cosine, or tangent from the side ratio.

Core Idea

SOH-CAH-TOA: $s i n 𝜃 = o p p o s i t e h y p o t e n u s e$ , $c o s 𝜃 = a d j a c e n t h y p o t e n u s e$ , $t a n 𝜃 = o p p o s i t e a d j a c e n t$ . Pick the ratio that connects the side you know to the side you need.

Understanding

SOH-CAH-TOA tells you which ratio connects the side you know to the side you need.

Sine uses opposite over hypotenuse.
Cosine uses adjacent over hypotenuse.
Tangent uses opposite over adjacent.

Label the sides from the angle in the problem, not from the angle you wish you had.

Step by Step

Identify the acute angle you'll work with.
Label the sides as opposite, adjacent, and hypotenuse relative to that angle.
Pick the trig ratio that connects the known side to the unknown side.
Set up the equation and solve: $u n k n o w n = k n o w n \times (t r i g v a l u e)$ or $u n k n o w n = k n o w n t r i g v a l u e$ .

Misconceptions

Labeling sides relative to the wrong angle — always use the angle named in the problem.
Using sine when you need cosine (or vice versa) because the sides are mislabeled.
Forgetting that the hypotenuse is always the longest side and is never "opposite" or "adjacent" in the ratio denominator context — it's always in a specific position.

Question

Worked Example

In right triangle $𝑃 𝑄 𝑅$ , angle $𝑃$ is a right angle, $s i n 𝑄 = 45$ , and $𝑃 𝑅 = 12$ . What is the length of $𝑄 𝑅$ ?

Select an answer to see the explanation