Question

Use an Addition orSubtraction Formula to find the exact value of the expression, asDemonstrated.

$\sin \left(-\frac{5\pi }{12}\right)$

Short answer

Therefore, the exact value of the expression is$-\frac{\sqrt{6}+\sqrt{2}}{4}$

Step-by-step solution

Step 1. Given information.

The given function is,$\sin \left(-\frac{5\pi }{12}\right)$.

Find the exact value of the expression.

Then determine the expression to get the answer.

Step 2. Determining the value.

Use the sum formula for the sine function where

$\sin (a+b)=\sin a\cos b+\cos a\sin b$

Therefore, $\sin \left(-\frac{5\pi }{12}\right)=-\sin \left(\frac{3\pi }{12}+\frac{2\pi }{12}\right)$

$=-\sin \left(\frac{3\pi }{12}+\frac{2\pi }{12}\right)$

$=-\left(\sin \frac{3\pi }{12}\cos \frac{2\pi }{12}+\cos \frac{3\pi }{12}\sin \frac{2\pi }{12}\right)$ the sum formula

$=-\left(\sin \frac{\pi }{4}\cos \frac{\pi }{6}+\cos \frac{\pi }{4}\sin \frac{\pi }{6}\right)$

$=-\left(\frac{\sqrt{2}}{2}\times \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times \frac{1}{2}\right)$

$=-\left(\frac{\sqrt{6}}{4}+\frac{\sqrt{2}}{4}\right)=-\frac{\sqrt{6}+\sqrt{2}}{4}$