Question

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

$\cos \frac{17\pi }{12}$

Short answer

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

Step-by-step solution

Step 1. Given information.

The given function is$\cos \frac{17\pi }{12}$

Step 2. Determining the value.

To evaluate $\cos \frac{17\pi }{12}$we can use the angle addition formula for cosine:

$\cos (x+y)=\cos x\cos y-\sin x\sin y$

Since $\frac{17\pi }{12}=\frac{15\pi }{12}+\frac{2\pi }{12}$

$=\frac{5\pi }{4}+\frac{\pi }{6}$

$x=\frac{5\pi }{4}$

And $y=\frac{\pi }{6}$

Therefore,

$\cos \frac{17\pi }{12}=\cos \left(\frac{5\pi }{4}+\frac{\pi }{6}\right)$

$=\cos \frac{5\pi }{4}\cos \frac{\pi }{6}-\sin \frac{5\pi }{4}\sin \frac{\pi }{6}$

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