Question

find the exact value of the expression

sin \frac { pi } { 12 } cos \frac { 2 pi } { 3 } + cos \frac { pi } { 12 } sin \frac { 2 pi } { 3 }

sin \frac { pi } { 12 } cos \frac { 2 pi } { 3 } + cos \frac { pi } { 12 } sin \frac { 2 pi } { 3 } = square
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Use the sine addition formula

The formula $\sin(A + B)=\sin A\cos B+\cos A\sin B$. Here $A=\frac{\pi}{12}$ and $B = \frac{2\pi}{3}$. So, $\sin\frac{\pi}{12}\cos\frac{2\pi}{3}+\cos\frac{\pi}{12}\sin\frac{2\pi}{3}=\sin(\frac{\pi}{12}+\frac{2\pi}{3})$.

Step2: Calculate the sum of angles

First, find a common denominator for the angles: $\frac{\pi}{12}+\frac{2\pi}{3}=\frac{\pi + 8\pi}{12}=\frac{9\pi}{12}=\frac{3\pi}{4}$.

Step3: Evaluate the sine function

We know that $\sin\frac{3\pi}{4}=\sin(\pi-\frac{\pi}{4})$. Using the identity $\sin(\pi - x)=\sin x$, so $\sin\frac{3\pi}{4}=\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$.

Answer:

$\frac{\sqrt{2}}{2}$