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 $90^{\circ}$ 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 $90^{\circ}$ 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

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

Say it as SOH-CAH-TOA. Label the sides first, then pick the ratio.

The values that show up again and again

Six numbers. Learning them means some questions take no work at all.

• $\sin 30^{\circ} = \frac{1}{2}$
• $\cos 60^{\circ} = \frac{1}{2}$
• $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$ and $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$
• $\tan 45^{\circ} = 1$
• $\sin 90^{\circ} = 1$ and $\sin 0^{\circ} = 0$

The $\sin 30^{\circ} = \frac{1}{2}$ one is worth extra attention. It says the side across from a $30^{\circ}$ angle is exactly half the hypotenuse. That turns some problems into a single division by $2$.

One identity

$\sin^{2}\theta + \cos^{2}\theta = 1$

This is true for every angle, always. The little $2$ means "square the whole thing," so $\sin^{2}\theta$ means $\sin\theta$ times $\sin\theta$. When a question hands you a sine and a cosine added like this, the answer is $1$.

Radians, in one line

Angles can be measured in degrees or in radians. A full circle is $360^{\circ}$, which is the same as $2\pi$ radians. To go from degrees to radians, multiply by $\frac{\pi}{180}$.