Trig Functions

, , from two pictures — right-triangle (SOH-CAH-TOA) and unit-circle ( θ, θ).

The idea

The trigonometric functions $\sin$, $\cos$ and $\tan$ each take an angle and return a number. They are well defined because in a right triangle the ratios of the sides depend only on the angle: two right triangles sharing an acute angle $\theta$ are similar, so each ratio of two sides comes out the same in both.

Name the sides relative to $\theta$: the opposite leg $a$ lies across from $\theta$, the adjacent leg $b$ lies beside it, and the hypotenuse $c$ is opposite the right angle. Then $\sin\theta = \frac{a}{c}, \qquad \cos\theta = \frac{b}{c}, \qquad \tan\theta = \frac{a}{b},$ and the mnemonic SOH-CAH-TOA records which ratio is which.

A right triangle cannot contain an angle of $90^{\circ}$ or more, so a second definition extends the first to every angle. Draw the unit circle, the circle of radius $1$ centered at the origin, and rotate a ray from the positive $x$-axis counterclockwise through the angle $\theta$. The ray meets the circle at the point $(\cos\theta, \sin\theta)$. For an acute angle the two definitions agree, since the radius is a hypotenuse of length $1$. But the ray can turn past a quarter turn, past a full turn, or clockwise for a negative angle, so the coordinates define $\cos\theta$ and $\sin\theta$ for every real $\theta$ — now taking negative values as well.

Ways to work on it

• Walkthrough. Both definitions, with a worked right-triangle example and the unit-circle picture.
• Practice. Compute , , for a random Pythagorean right triangle.
• Hardest. Mixed drill — right-triangle (SOH-CAH-TOA) values and unit-circle coordinates, including obtuse and negative angles.

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