Exact trigonometric values

#### Tier: #Higher

Description

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 $90°$.

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