For certain angles, you are expected to know exact values of sin, cos and tan without a calculator.
| Angle | $\sin$ | $\cos$ | $\tan$ |
|---|---|---|---|
| $0°$ | $0$ | $1$ | $0$ |
| $30°$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
| $45°$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ |
| $60°$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
| $90°$ | $1$ | $0$ | undefined |
These come from two special right-angled triangles:
- 30-60-90 triangle: sides $1$, $\sqrt{3}$, $2$
- 45-45-90 triangle: sides $1$, $1$, $\sqrt{2}$
$$\cos 60° = \frac{1}{2}, \quad \sin 30° = \frac{1}{2}, \quad \tan 45° = 1$$
Notice that $\sin$ and $\cos$ swap between $30°$ and $60°$ — a useful check.
Common error: mixing up $\sin 30°$ and $\cos 30°$. Memorise by noting sine increases ($0 \to \frac{1}{2} \to \frac{\sqrt{2}}{2} \to \frac{\sqrt{3}}{2} \to 1$) as angle increases from $0°$ to $90°$.