use a half - angle formula to find the exact value of $\\cos\\frac{11\\pi}{12}$. $\\cos\\frac{11\\pi}{12}=\\s…

Explanation:

Step1: Choose the half - angle formula

The half - angle formula for cosine is $\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}$. Since $\frac{11\pi}{12}=\frac{\frac{11\pi}{6}}{2}$, and $\frac{11\pi}{12}$ is in the second quadrant where cosine is negative.

Step2: Find the value of $\cos\frac{11\pi}{6}$

We know that $\cos\frac{11\pi}{6}=\cos(2\pi-\frac{\pi}{6})=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$.

Step3: Substitute into the half - angle formula

Substitute $\alpha=\frac{11\pi}{6}$ into $\cos\frac{\alpha}{2}=-\sqrt{\frac{1+\cos\alpha}{2}}$ (negative because $\frac{11\pi}{12}$ is in the second quadrant). [ \begin{align*} \cos\frac{11\pi}{12}&=-\sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}}\ &=-\sqrt{\frac{2+\sqrt{3}}{4}}\ &=-\frac{\sqrt{2+\sqrt{3}}}{2} \end{align*} ]

Answer:

$-\frac{\sqrt{2+\sqrt{3}}}{2}$