Trig Values

, , at 0, π/6, π/4, π/3, π/2 — the unit circle by heart.

The idea

The special angles $0$, $\pi/6$, $\pi/4$, $\pi/3$ and $\pi/2$ are the angles whose sine, cosine and tangent come out as exact fractions and square roots rather than calculator decimals. These are the values to know by heart.

Two triangles produce all the middle entries. Cut an equilateral triangle of side $2$ down the middle: the half is a right triangle with angles $30^{\circ}$ and $60^{\circ}$ — that is, $\pi/6$ and $\pi/3$ — and sides $1$, $\sqrt{3}$, $2$. Cut a unit square along its diagonal: the half has two angles of $45^{\circ}$, or $\pi/4$, and sides $1$, $1$, $\sqrt{2}$. Reading side ratios off these two triangles recovers every value at $\pi/6$, $\pi/4$ and $\pi/3$.

The two ends come from the unit circle. At $\theta = 0$ the ray along the positive $x$-axis meets the circle at $(1, 0)$; at $\theta = \pi/2$ it points straight up, at $(0, 1)$. Cosine is the first coordinate and sine the second, so these values read off directly — and $\tan\theta = \sin\theta/\cos\theta$ is undefined at $\pi/2$, where the cosine is $0$.

As $\theta$ grows from $0$ to $\pi/2$, the sine climbs through the same values the cosine descends through.

Ways to work on it

Not sure where to start? Take the ten-question placement test.