Trigonometry — sin · cos

Memory Point

The Golden Mnemonic

Memorize one word. Unlock all three ratios forever.

The master key

SOH · CAH · TOA

SOH — Sine = Opposite / Hypotenuse · CAH — Cosine = Adjacent / Hypotenuse · TOA — Tangent = Opposite / Adjacent

sin θ

⚡ "SINE = OPPOSITE SIDE"

\( \sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}} \)

\( \sin\theta = \dfrac{y}{r} \) on unit circle

sin 0° = 0 sin 30° = ½ sin 45° = √2/2 sin 60° = √3/2 sin 90° = 1

cos θ

⚡ "COSINE = CLOSE (ADJACENT)"

\( \cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}} \)

\( \cos\theta = \dfrac{x}{r} \) on unit circle

cos 0° = 1 cos 30° = √3/2 cos 45° = √2/2 cos 60° = ½ cos 90° = 0

tan θ

⚡ "TAN = TOP over ADJACENT"

\( \tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}} = \dfrac{\sin\theta}{\cos\theta} \)

\( \tan\theta = \dfrac{y}{x} \) on unit circle

tan 0° = 0 tan 30° = 1/√3 tan 45° = 1 tan 60° = √3 tan 90° = ∞

Quick Reference

Key Angle Values

The 5 angles you must know cold.

Degrees	Radians	sin θ	cos θ	tan θ
0°	\(0\)	\(0\)	\(1\)	\(0\)
30°	\(\frac{\pi}{6}\)	\(\frac{1}{2}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{3}}\)
45°	\(\frac{\pi}{4}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(1\)
60°	\(\frac{\pi}{3}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{2}\)	\(\sqrt{3}\)
90°	\(\frac{\pi}{2}\)	\(1\)	\(0\)	undefined

sin · cos · tan