AHL 3.11 (HL)—Further trigonometric equations

Supplementary angles reveal trigonometric graph symmetry.

The AHL 3.11 identities are $\sin(\pi-\theta)=\sin\theta$, $\cos(\pi-\theta)=-\cos\theta$, and $\tan(\pi-\theta)=-\tan\theta$. They follow from the unit circle: supplementary angles have the same vertical coordinate and opposite horizontal coordinates.

Example

If $\theta=\pi/6$, then $\sin(5\pi/6)=1/2$, $\cos(5\pi/6)=-\sqrt3/2$, and $\tan(5\pi/6)=-1/\sqrt3$.

Locate $\pi-\theta$ in quadrant II, then keep the sine sign and reverse the cosine and tangent signs.

The identity describes supplementary-angle symmetry; it does not say that every trigonometric function keeps its value under $\theta\mapsto\pi-\theta$.