Convention

Exact trigonometric ratios of the standard angles

Trigonometry • Trigonometric Identity

Statement

sin 0^{\circ}, sin 3 0^{\circ}, sin 4 5^{\circ}, sin 6 0^{\circ}, sin 9 0^{\circ}

are

0, \frac{1}{2}, \frac{1}{2}, \frac{3}{2}, 1

; the cosines are the same list reversed; and

tan 3 0^{\circ} = \frac{1}{3}

tan 4 5^{\circ} = 1

tan 6 0^{\circ} = 3

How to use it

1Recover the values from two triangles rather than memorising a table: the half-equilateral gives $3 0^{\circ}$ and $6 0^{\circ}$ with sides $1, 3, 2$ , and the half-square gives $4 5^{\circ}$ with sides $1, 1, 2$ .
2Sine increases and cosine decreases across $0^{\circ}$ to $9 0^{\circ}$ , which catches a swapped pair at a glance.
3Leave the surds unevaluated when the printed options are surds; decimals throw away the exactness the answer needs.

Evaluate

sin 6 0^{\circ} cos 3 0^{\circ} + sin 3 0^{\circ} cos 6 0^{\circ}

Step-by-step

1
$= \frac{3}{2} \times \frac{3}{2} + \frac{1}{2} \times \frac{1}{2}$
Why: Substitute the exact values rather than decimals.
2
$= \frac{3}{4} + \frac{1}{4} = 1$

Answer

1

Watch out

The commonest slip is interchanging

\frac{3}{2}

and

\frac{1}{2}

, that is using the

4 5^{\circ}

value where

6 0^{\circ}

belongs.

tan 9 0^{\circ}

is undefined, not infinite in any usable sense, and a ratio taken of the wrong acute angle silently swaps sine with cosine.

Also known as

exact trig values, standard angles, sin 30 cos 60, trigonometric table, special angles