Special angles and reference angles | Trigonometry

The special angles

Three angles have exact values that come from geometry.

$$45°$$ : the terminal side is the line $$y = x$$ , so the point on the unit circle has equal coordinates. From $$2x^2 = 1$$ , both are $\frac{\sqrt{2}}{2}$ .
$$30°$$ and $$60°$$ : cut an equilateral triangle with side $$1$$ in half. You get a 30-60-90 triangle with hypotenuse $$1$$ , short leg $\frac{1}{2}$ and long leg $\frac{\sqrt{3}}{2}$ .

RuleExact values

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

The point on the unit circle for $\theta$ is a mirror image of the point for its reference angle. The coordinates have the same absolute values, so:

Reflecting a point across the $$x$$ -axis turns $\theta$ into $-\theta$ : the $$x$$ -coordinate stays, the $$y$$ -coordinate changes sign.

Quick check

What is $\cos 60°$ ?

Quick check

What is $\sin 300°$ ?

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