20.4 Exact trigonometric values — Learn

Prerequisites: special right-angled triangles, surds, and exact fractions.

Objectives: give exact sine and cosine values for $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$, and exact tangent values for $0^\circ$, $30^\circ$, $45^\circ$, and $60^\circ$.

An exact value uses fractions and surds instead of a rounded decimal. A surd is a root kept in exact form, such as $\sqrt2$. A $45^\circ$-$45^\circ$-$90^\circ$ triangle has sides $1,1,\sqrt2$, so $\sin45^\circ=\cos45^\circ=\frac{\sqrt2}{2}$ and $\tan45^\circ=1$. A $30^\circ$-$60^\circ$-$90^\circ$ triangle has sides $1,\sqrt3,2$.

Example 1. Work out $2\sin30^\circ+\cos60^\circ$ exactly.

Example 2. Work out $\tan60^\circ\times\tan30^\circ$ exactly.

Watch out. $\tan90^\circ$ is undefined because its cosine denominator is zero. Do not copy a large calculator value as an exact answer.

Progress check

The garden diagonal can be found exactly if its sides are chosen to form a special triangle. Which side lengths would make that possible?

You should now be able to

• give the exact sine and cosine values for the five special angles;
• give the exact tangent values for the four defined special angles;
• simplify expressions using exact values, and explain why $\tan90^\circ$ is undefined.