use an addition or subtraction formula to find the exact value of the expression, as demonstrated in example…

Explanation:

Step1: Rewrite the angle

We can rewrite $\frac{43\pi}{12}$ as $\frac{36\pi + 7\pi}{12}=3\pi+\frac{7\pi}{12}$. Since $\sin(x + 3\pi)=-\sin(x)$, then $\sin(\frac{43\pi}{12})=-\sin(\frac{7\pi}{12})$. And $\frac{7\pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}$.

Step2: Apply the sine - addition formula

The sine - addition formula is $\sin(A + B)=\sin A\cos B+\cos A\sin B$. Here $A=\frac{\pi}{3}$ and $B = \frac{\pi}{4}$. So $\sin(\frac{\pi}{3}+\frac{\pi}{4})=\sin\frac{\pi}{3}\cos\frac{\pi}{4}+\cos\frac{\pi}{3}\sin\frac{\pi}{4}$. We know that $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$, $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, $\cos\frac{\pi}{3}=\frac{1}{2}$, $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. Then $\sin(\frac{\pi}{3}+\frac{\pi}{4})=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}$.

Step3: Get the final result

Since $\sin(\frac{43\pi}{12})=-\sin(\frac{7\pi}{12})$, then $\sin(\frac{43\pi}{12})=-\frac{\sqrt{6}+\sqrt{2}}{4}$.

Answer:

$-\frac{\sqrt{6}+\sqrt{2}}{4}$