Trigonometry — Core Skills

Trigonometry: The Basics

Part 1 of 2 — One Skill, One Idea

Trigonometry on the SAT starts with one shape: a right triangle, which is a triangle with one $90°$ angle.

Naming the three sides

Pick one of the two slanted angles and call it $\theta$ (that is the Greek letter theta — it is a name for "the angle we care about"). Now the three sides get names:

• Hypotenuse — the longest side, always the one across from the $90°$ angle. Its name never changes.
• Opposite — the side across the triangle from $\theta$.
• Adjacent — the other leg, the one touching $\theta$.

Opposite and adjacent swap places if you pick the other angle. That is normal. Find $\theta$ first, then label.

SOH-CAH-TOA

Three ratios, and this word helps you remember all three:

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

Sine is Opposite over Hypotenuse. Cosine is Adjacent over Hypotenuse. Tangent is Opposite over Adjacent.

Worked example

Picture a right triangle. The angle $\theta$ sits at the bottom left. The side going straight up across from $\theta$ has length $3$. The bottom side touching $\theta$ has length $4$. The slanted side across from the right angle has length $5$.

Label them: opposite $= 3$, adjacent $= 4$, hypotenuse $= 5$.

Now read the ratios straight off:

$\sin\theta = \frac{3}{5} \qquad \cos\theta = \frac{4}{5} \qquad \tan\theta = \frac{3}{4}$

That is the whole move. Label the three sides, then pick the ratio the question asks for.

Trig Values Worth Memorizing

Part 2 of 2 — Practice

The three ratios from Part 1

• $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$