Exact trigonometric values and trigonometric graphs

Last time

Before we start: last time, in Trigonometric ratios and angles of elevation, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

A wave that repeats

Sound, light and ocean waves all repeat. The sine curve is the simplest wave: it rises to one, falls to minus one, and repeats every three hundred and sixty degrees. Some angles give exact values you can write without a calculator. Learn those, learn the three curves, and a whole group of questions becomes easy.

Exact values

The special angles are zero, thirty, forty-five, sixty and ninety. Sine goes zero, a half, root two over two, root three over two, one. Cosine is the same list backwards: one, root three over two, root two over two, a half, zero. Tangent is sine over cosine: zero, one over root three, one, root three, and undefined at ninety.

Where the values come from

An equilateral triangle of side two, cut in half, gives a right triangle with sides one, root three and two, and angles thirty and sixty. A square with side one has a diagonal of root two, which gives a forty-five forty-five ninety triangle. Read off sine, cosine and tangent from those sides. You can rebuild the table in thirty seconds.