Trigonometry Reference
Unit Circle
On the unit circle, a point has coordinates $(\cos\theta,\sin\theta)$. This page keeps the core values visible without trying to become the full interactive lab yet.
Fact Table
| Quadrant | Angle Range | Signs |
|---|---|---|
| I | $0^\circ$ to $90^\circ$ | $\sin\theta>0,\ \cos\theta>0,\ \tan\theta>0$ |
| II | $90^\circ$ to $180^\circ$ | $\sin\theta>0,\ \cos\theta<0,\ \tan\theta<0$ |
| III | $180^\circ$ to $270^\circ$ | $\sin\theta<0,\ \cos\theta<0,\ \tan\theta>0$ |
| IV | $270^\circ$ to $360^\circ$ | $\sin\theta<0,\ \cos\theta>0,\ \tan\theta<0$ |
Content Formulas
Unit Circle Point
$$(x,y)=(\cos\theta,\sin\theta)$$
Tangent
$$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac yx,\quad x\ne0$$
Remember the first-quadrant reference values, then use quadrant signs to place them around the circle.
Common Values
| Degrees | Radians | $\cos\theta$ | $\sin\theta$ |
|---|---|---|---|
| $0^\circ$ | $0$ | $1$ | $0$ |
| $30^\circ$ | $\frac{\pi}{6}$ | $\frac{\sqrt3}{2}$ | $\frac12$ |
| $45^\circ$ | $\frac{\pi}{4}$ | $\frac{\sqrt2}{2}$ | $\frac{\sqrt2}{2}$ |
| $60^\circ$ | $\frac{\pi}{3}$ | $\frac12$ | $\frac{\sqrt3}{2}$ |
| $90^\circ$ | $\frac{\pi}{2}$ | $0$ | $1$ |
Classic Examples
Coordinates from an Angle
Find the unit-circle point for $150^\circ$.
Solution StrategyThe reference angle is $30^\circ$, and $150^\circ$ is in Quadrant II.
$$150^\circ=180^\circ-30^\circ$$
$$\cos150^\circ=-\frac{\sqrt3}{2}$$
$$\sin150^\circ=\frac12$$
$$\left(-\frac{\sqrt3}{2},\frac12\right)$$
Find Tangent
Find $\tan(225^\circ)$.
Solution StrategyThe reference angle is $45^\circ$, and tangent is positive in Quadrant III.
$$225^\circ=180^\circ+45^\circ$$
$$\tan(225^\circ)=1$$