Exact Values & the Unit Circle

On the non-calculator Paper 1, $\sin 60^\circ$ is not a button, it’s $\dfrac{\sqrt{3}}{2}$, instantly. This subtopic is pure fluency, built from two triangles and one circle.

The table (degrees and radians)

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

($\frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$; both forms accepted.) The radian labels must be as fluent as the degree ones.

The two regenerating triangles

Memory fails under pressure; derivation doesn’t. $45^\circ$: a right isosceles triangle with legs $1$, hypotenuse $\sqrt{2}$, gives $\sin 45^\circ = \cos 45^\circ = \dfrac{1}{\sqrt{2}}$, $\tan 45^\circ = 1$. $30^\circ/60^\circ$: an equilateral triangle of side $2$, halved, base $\frac{1}{2}\times 2 = 1$, height $\sqrt{3}$, gives the whole $30^\circ/60^\circ$ column. Sketching a triangle takes ten seconds and rescues a blanked value mid-exam.

The unit circle: exact values everywhere

A point at angle $\theta$ on the circle of radius $1$ sits at $(\cos\theta, \sin\theta)$. From that one picture:

• Quadrant signs, the CAST pattern, read off the coordinates’ signs rather than memorised
• Reference angles, $\sin 150^\circ = \sin 30^\circ = \frac{1}{2}$ (same height, second quadrant); $\cos 225^\circ = -\cos 45^\circ = -\dfrac{1}{\sqrt{2}}$
• The symmetries behind multi-solution trig equations: $\sin(180^\circ - \theta) = \sin\theta$, $\cos(360^\circ - \theta) = \cos\theta$, $\tan(180^\circ + \theta) = \tan\theta$

Find $\tan 240^\circ$ exactly: $240^\circ$ is third quadrant ($\tan$ positive), reference angle $60^\circ$ $\to$ $\tan 240^\circ = \sqrt{3}$

The routine: quadrant $\to$ sign $\to$ reference angle $\to$ table value. Four steps, written as one line of working.

Where this fluency gets spent

Everywhere: trig equations on Paper 1, exact sector-area triangles, exact values inside calculus (evaluating derivatives at $\frac{\pi}{3}$), surd arithmetic feeding non-calculator speed. It’s a small table with the highest reuse rate in the syllabus.

Common mistakes

• $\sin$ and $\cos$ columns swapped at $30^\circ/60^\circ$
• Quadrant signs guessed instead of read from the circle
• Radian-labelled angles converted to degrees “to be safe” (slow, and invites range errors)
• $\tan 90^\circ$ given a value (undefined, say so)
• $\frac{1}{\sqrt{2}}$ “simplified” incorrectly when rationalising

Full topic context: Trigonometry notes.