Signs in the four quadrants
The coordinate plane is divided into four quadrants. The sign of $\sin$, $\cos$ and $\tan$ depends on the quadrant of the angle $\theta$ (measured anticlockwise from the positive $x$-axis):
- Quadrant I ($0^\circ$–$90^\circ$): all positive;
- Quadrant II ($90^\circ$–$180^\circ$): only $\sin$ positive;
- Quadrant III ($180^\circ$–$270^\circ$): only $\tan$ positive;
- Quadrant IV ($270^\circ$–$360^\circ$): only $\cos$ positive.
Reference angle
The reference angle is the acute angle between the terminal side and the $x$-axis. We find the value of the ratio for the reference angle, then attach the correct sign for the quadrant.
Key formula
$\text{Q II: }180^\circ-\theta,\quad \text{Q III: }\theta-180^\circ,\quad \text{Q IV: }360^\circ-\theta$
Special angles
Memorise $\sin 30^\circ=\tfrac{1}{2}$, $\cos 30^\circ=\tfrac{\sqrt3}{2}$, $\tan 30^\circ=\tfrac{1}{\sqrt3}$, $\sin 45^\circ=\cos 45^\circ=\tfrac{1}{\sqrt2}$, $\tan 45^\circ=1$, $\sin 60^\circ=\tfrac{\sqrt3}{2}$, $\cos 60^\circ=\tfrac{1}{2}$.
Example
$\cos 150^\circ$: the angle is in Quadrant II (cosine negative) with reference angle $180^\circ-150^\circ=30^\circ$. So $\cos 150^\circ=-\cos 30^\circ=-\tfrac{\sqrt3}{2}$.
Remember
"All, Sin, Tan, Cos" tells you which ratio is positive in Quadrants I, II, III, IV.