what is the exact value of sin(7π/12)?\n√6 - √2 / 8\n√6 - √2 / 4\n√6 + √2 / 8\n√6 + √2 / 4

Explanation:

Step1: Rewrite the angle

We know that $\frac{7\pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}$. Then, by the sum - formula for sine $\sin(A + B)=\sin A\cos B+\cos A\sin B$, where $A=\frac{\pi}{3}$ and $B = \frac{\pi}{4}$.

Step2: Find the values of trigonometric functions

We know that $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$, $\cos\frac{\pi}{3}=\frac{1}{2}$, $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$.

Step3: Apply the sum - formula

$\sin(\frac{\pi}{3}+\frac{\pi}{4})=\sin\frac{\pi}{3}\cos\frac{\pi}{4}+\cos\frac{\pi}{3}\sin\frac{\pi}{4}=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}$.

Answer:

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