Answer
The reduction formula is $$-\sin\theta$$
Work Step by Step
*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
$$-\sin\theta$$
