Exact Values

Video lesson · from 4:56

One lesson video covers all of Trigonometry, so it opens at 4:56 for this topic — not from the beginning.

It is important to know the exact values for the trigonometric functions at key angles.

Degrees	Radians	$\sin\theta$	$\cos\theta$	$\tan\theta$
$0^\circ$	$0$	$0$	$1$	$0$
$90^\circ$	$\frac{\pi}{2}$	$1$	$0$	undefined
$180^\circ$	$\pi$	$0$	$-1$	$0$
$270^\circ$	$\frac{3\pi}{2}$	$-1$	$0$	undefined
$360^\circ$	$2\pi$	$0$	$1$	$0$

You can easily remember these values using two special right-angled triangles.

For $45^\circ$ (or $\frac{\pi}{4}$ )

For $30^\circ$ and $60^\circ$ (or $\frac{\pi}{6}$ and $\frac{\pi}{3}$ )

Function	$30^\circ \quad \left(\frac{\pi}{6}\right)$	$45^\circ \quad \left(\frac{\pi}{4}\right)$	$60^\circ \quad \left(\frac{\pi}{3}\right)$
$\sin\theta$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$
$\cos\theta$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$
$\tan\theta$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$

These must be memorised: exact values are a known weak spot across the course, and in the non-calculator paper there is no way round them. Learn the two triangles rather than the table — you can rebuild every value from them.
Exact means exact: a decimal such as 0.866 gains nothing when $\frac{\sqrt{3}}{2}$ was asked for.
Rationalise the denominator: $\frac{1}{\sqrt{2}}$ is usually written $\frac{\sqrt{2}}{2}$ . Check the form expected.
Know them in radians as well as degrees: the same values appear as $\frac{\pi}{6}$ , $\frac{\pi}{4}$ and $\frac{\pi}{3}$ , and questions mix the two freely.