use a sum or difference formula to find an exact value of $\\sin(\\frac{7\\pi}{12})$

Explanation:

Step1: Express $\frac{7\pi}{12}$ as sum of angles

We know that $\frac{7\pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}$.

Step2: Use the sine sum formula

The sine sum formula is $\sin(A + B)=\sin A\cos B+\cos A\sin B$. Here $A=\frac{\pi}{3}$ and $B = \frac{\pi}{4}$. We know that $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$, $\cos\frac{\pi}{3}=\frac{1}{2}$, $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. Substitute these values into the formula: [ \begin{align*} \sin\left(\frac{7\pi}{12}\right)&=\sin\left(\frac{\pi}{3}+\frac{\pi}{4}\right)\ &=\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} \end{align*} ]

Answer:

$\frac{\sqrt{2}+\sqrt{6}}{4}$ (the third option)