# SAT right triangles and trig: SOH-CAH-TOA is just three ratios

> Right-triangle questions run on the Pythagorean theorem, three trig ratios, and two special triangles. Plus the complement identity the SAT loves to test.

Published 2026-05-06 | Updated 2026-08-01 | Math | StudyHall

Canonical: https://trystudyhall.com/blog/sat-right-triangles-and-trigonometry

Trigonometry has a reputation, and the SAT does not deserve credit for it. SAT trig stays inside right triangles: one theorem, three ratios, two special triangles, one heavily tested identity. None of it is magic. Sine, cosine, and tangent are names for side ratios, and once you label the sides relative to the angle, every question becomes a fraction you can write down.

SAT trigonometry is right-triangle trig only: the Pythagorean theorem, the sine, cosine, and tangent ratios (SOH-CAH-TOA), the two special triangles from the reference sheet, and the identity that sine of an angle equals cosine of its complement. The unit circle and trig graphs are not tested, so the whole topic fits on one page.

**Key takeaways:**
- **Pythagorean theorem:** $a^2 + b^2 = c^2$, where $c$ is the hypotenuse, always the side opposite the right angle.
- Recognize the **families**: 3-4-5 and 5-12-13, plus their multiples like 6-8-10 and 10-24-26.
- **SOH-CAH-TOA:** $\sin = \tfrac{\text{opp}}{\text{hyp}}$, $\cos = \tfrac{\text{adj}}{\text{hyp}}$, $\tan = \tfrac{\text{opp}}{\text{adj}}$. Ratios, fixed by the angle.
- **The identity:** $\sin(x^\circ) = \cos(90^\circ - x^\circ)$. When $\sin$ of one angle equals $\cos$ of another, the angles sum to $90$.

## The Pythagorean theorem and the triangles worth recognizing

In any right triangle, the legs and hypotenuse obey $a^2 + b^2 = c^2$. The SAT reuses a few integer triangles constantly: **3-4-5** and **5-12-13**, their scaled copies (6-8-10, 10-24-26), and the occasional 8-15-17. Recognizing a family turns a square-root computation into a lookup. See legs of 5 and 12, write 13, move on. For the angle relationships that sit beside these side rules, see [angles and triangles](https://trystudyhall.com/blog/sat-angles-and-triangles).

> **Watch out: The hypotenuse is a position, not a guess** The hypotenuse is the side opposite the right angle, full stop (also always the longest side). Given sides $6$ and $10$ where $10$ is the hypotenuse, the missing leg is $\sqrt{100 - 36} = 8$, not $\sqrt{136}$. Plugging the hypotenuse in as a leg is the most common Pythagorean error.

## SOH-CAH-TOA: three fractions with names

For an acute angle in a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. The deep fact is that these ratios depend only on the angle: every right triangle with that angle is a scaled copy of the same shape, so the fractions never change. The practical move is to label first. Mark the angle you care about, then write opp, adj, and hyp on the sides *before* touching the ratio.

*Figure: An 8-15-17 right triangle with the right angle at C. Relative to angle A: opposite BC = 8, adjacent AC = 15, hypotenuse AB = 17.*

## Try one

**Example: Right triangles and trigonometry.**

In right triangle $ABC$, the right angle is at $C$, $AC = 15$, $BC = 8$, and $AB = 17$. What is the value of $\tan A$?

- A) $\tfrac{8}{17}$
- B) $\tfrac{15}{8}$
- C) $\tfrac{8}{15}$
- D) $\tfrac{15}{17}$

**Answer:** C. Label relative to angle $A$: opposite $BC = 8$, adjacent $AC = 15$, hypotenuse $AB = 17$. Tangent is opposite over adjacent, so $\tan A = \tfrac{8}{15}$. The trap $\tfrac{15}{8}$ inverts the ratio (that is $\tan B$); $\tfrac{8}{17}$ is $\sin A$; $\tfrac{15}{17}$ is $\cos A$. Labeling first makes the four choices stop looking interchangeable.

## The special triangles (they are on the reference sheet)

Two triangles come with fixed side ratios, and both are drawn on [the digital SAT’s reference sheet](https://trystudyhall.com/blog/sat-geometry-formulas), so this is recognition, not recall. A **45-45-90** triangle has legs $x$ and hypotenuse $x\sqrt{2}$. A **30-60-90** triangle has sides $x$, $x\sqrt{3}$, and $2x$, shortest side opposite the $30^\circ$ angle. The one skill the sheet cannot supply is mapping: given one side, identify which slot it fills, then scale the other two. Most misses come from assigning $x\sqrt{3}$ to the wrong side, not from forgetting the pattern.

## Sine of an angle, cosine of its complement

The two acute angles of a right triangle add to $90^\circ$, and the side opposite one is the side adjacent to the other. So $\sin A = \cos B$ whenever $A + B = 90^\circ$. If a question says $\sin(x^\circ) = \cos(y^\circ)$, the entire content of the question is $x + y = 90$. No triangle needs drawing, no calculator helps. (Angles here stay in degrees; radians live over in [circle questions](https://trystudyhall.com/blog/sat-circles).)

**Example: Right triangles and trigonometry.**

If $\sin(x^\circ) = \cos(38^\circ)$ and $0 < x < 90$, what is the value of $x$?

- A) $38$
- B) $52$
- C) $62$
- D) $128$

**Answer:** B. Sine of an angle equals cosine of its complement, so $x + 38 = 90$ and $x = 52$. The trap $38$ assumes sine and cosine agree at the same angle (they do only at $45^\circ$); $62$ subtracts from $100$ instead of $90$; $128$ adds $38$ to $90$ instead of subtracting, and also lands outside the allowed range, which is its own warning sign.

**[Practice Right Triangles & Trig](https://trystudyhall.com/learn/right-triangles-and-trigonometry)**: Drill the ratios, the families, and the complement identity with a tutor that makes you label the sides out loud.

## FAQ

### What trigonometry do you actually need for the SAT?

Right-triangle trig only: the Pythagorean theorem, the sine, cosine, and tangent ratios, the two special triangles, and the identity that sine of an angle equals cosine of its complement. The unit circle and trig graphs are not tested.

### What are the special right triangles on the SAT?

45-45-90, with legs equal and hypotenuse leg-times-root-2, and 30-60-90, with sides in the ratio x, x root 3, and 2x, shortest side opposite the 30 degree angle. Both are printed on the reference sheet, so the skill is mapping the given side to the right slot.

### Why does sin(x) equal cos(90 - x)?

The two acute angles in a right triangle sum to 90 degrees, and the side opposite one is the side adjacent to the other, so sine for one angle is the same fraction as cosine for its complement. On the SAT, sin(x) = cos(y) translates directly to x + y = 90.

### How do I know which side is the hypotenuse?

It is the side opposite the right angle, always the longest side. Identify it before using the Pythagorean theorem or any trig ratio, because plugging a leg into the hypotenuse slot is the most common error in this category.

[Read the HTML version](https://trystudyhall.com/blog/sat-right-triangles-and-trigonometry) | [Browse the StudyHall blog](https://trystudyhall.com/blog)