Reduction formulae

Place an angle $\theta$ in standard position: start on the positive $x$-axis and turn anticlockwise. A point $(x;y)$ on the terminal arm at distance $r$ from the origin gives

$\sin \theta =\frac{y}{r}\cos \theta =\frac{x}{r}\tan \theta =\frac{y}{x}$

Since $r$ is always positive, the signs of $x$ and $y$ decide the signs of the ratios:

• Quadrant 1 (0° to 90°): All positive
• Quadrant 2 (90° to 180°): Sin positive
• Quadrant 3 (180° to 270°): Tan positive
• Quadrant 4 (270° to 360°): Cos positive

Example. If $\tan \theta =-\frac{4}{3}$ and $90°<\theta <270°$, then $\theta$ is in quadrant 2. Take $x=-3$, $y=4$, $r=5$: $\sin \theta =\frac{4}{5}$ and $\cos \theta =-\frac{3}{5}$.

The ratio stays the same; only the sign changes, according to CAST.

• $180°-\theta$ (quadrant 2): $\sin \theta$, $-\cos \theta$, $-\tan \theta$
• $180°+\theta$ (quadrant 3): $-\sin \theta$, $-\cos \theta$, $\tan \theta$
• $360°-\theta$ (quadrant 4): $-\sin \theta$, $\cos \theta$, $-\tan \theta$
• $360°+\theta$: same as $\theta$

Examples.

• $\cos 160°=\cos (180°-20°)=-\cos 20°$
• $\tan 250°=\tan (180°+70°)=\tan 70°$
• $\sin 300°=\sin (360°-60°)=-\sin 60°$

For angles above 360°, subtract 360° until the angle is between 0° and 360°.

A negative angle is measured clockwise, which puts $-\theta$ in quadrant 4:

• $\sin (-\theta )=-\sin \theta$
• $\cos (-\theta )=\cos \theta$
• $\tan (-\theta )=-\tan \theta$

Example. $\cos (-240°)=\cos 240°=\cos (180°+60°)=-\cos 60°=-\frac{1}{2}$

Deal with the negative sign first, then reduce the angle.