Given the following expression:
[tex]cos(\frac{17\pi}{12})[/tex]
We will find the exact value of cosine the function
First, we will find the reference angle
[tex]\frac{17\pi}{12}=\frac{17\pi}{12}-\pi=\frac{5\pi}{12}[/tex]
The given angle lying in the third quadrant and has a reference angle of (5π/12)
We will find the cosine function using the sum of the cosine identity using the standard angles.
[tex]cos(\frac{5\pi}{12})=cos(\frac{\pi}{4}+\frac{\pi}{6})=cos(\frac{\pi}{4})cos(\frac{\pi}{6})-sin(\frac{\pi}{4})sin(\frac{\pi}{6})[/tex]
substitute the sines and cosines of the standard angles:
[tex]cos(\frac{5\pi}{12})=\frac{1}{\sqrt{2}}*\frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}*\frac{1}{2}=\frac{\sqrt{3}-1}{2\sqrt{2}}=\frac{\sqrt{6}-\sqrt{2}}{4}[/tex]
So, the exact value of the given function:
[tex]cos(\frac{17\pi}{12})=-cos(\frac{5\pi}{12})=\frac{\sqrt{2}-\sqrt{6}}{4}[/tex]
The answer will be option 3
[tex]\frac{\sqrt{2}-\sqrt{6}}{4}[/tex]
