Question

what is the value of sin(180+X)?

Answer 1

It is -sin x.

There are multiple of ways of checking it. You can mentally recall the graph of sin x, where $\sin x\&\sin \left(180°-x\right)$ have the same value on the positive x-axis, and $\sin \left(180°+x\right)$ is the mirror image of sin x, because $\sin \left(180°+x\right)$ has the same value as sin x but on the negative y-axis.

You can also use the formula $\sin \left(90°-x\right)=\cos x$ to evaluate its value mentally:-

$$\begin{gathered} \sin \left(180°+x\right) \\ =\sin \left[90°-\left(-90°-x\right)\right] \end{gathered}$$
$$\begin{gathered} =\cos \left(-90°-x\right) \\ =\cos \left(90°+x\right)\left[\because \cos (-x)=\cos x\right] \end{gathered}$$
$$\begin{gathered} =\cos \left[90°-\left(-x\right)\right] \\ =\sin \left(-x\right) \\ =-\sin x \end{gathered}$$