## Trigonometry (11th Edition) Clone

The reduction formula is $$-\sin\theta$$
*Summary of the method: For a formula $f(Q\pm\theta)$ 1) See that $Q$ terminates on the $x$ or $y$ axis. If it terminates on the $x$ axis, go for Case 1. If it terminates on the $y$ axis, go for Case 2. 2) Case 1: - For a small positive value of $\theta$, determinate $Q\pm\theta$ lies in which quadrant. - If $f\gt0$, use a $+$ sign. If $f\lt0$, use a $-$ sign. - The reduced form will have that sign, $f$ the function and $\theta$ the angle. 3) Case 2: - For a small positive value of $\theta$, determinate $Q\pm\theta$ lies in which quadrant. - If $f\gt0$, use a $+$ sign. If $f\lt0$, use a $-$ sign. - The reduced form will have that sign, cofunction of$f$ as the function and $\theta$ the angle. $$\sin(180^\circ+\theta)$$ 1) $180^\circ$ terminates on the $x$ axis. We go for Case 1. 2) As $\theta$ is a very small positive value, which means $\theta\gt0$, $$180^\circ\lt(180^\circ+\theta)\lt270^\circ$$ So $180^\circ+\theta$ lies in quadrant III. 3) Cosine is negative in quadrant III. So we use a $-$ sign. 4) In case 1, we use the same the given formula as the function, which here is sine, combined with the negative sign proved above. Overall, the reduced form would be $$-\sin\theta$$