find the exact value of the expression.\n cosleft(\frac{5pi}{4}-\frac{11pi}{6}\right)=square \n(type an…

Explanation:

Step1: Find a common - denominator for the angles

First, find a common denominator for $\frac{5\pi}{4}$ and $\frac{11\pi}{6}$. The common denominator of 4 and 6 is 12. So, $\frac{5\pi}{4}-\frac{11\pi}{6}=\frac{5\pi\times3}{4\times3}-\frac{11\pi\times2}{6\times2}=\frac{15\pi}{12}-\frac{22\pi}{12}=-\frac{7\pi}{12}$.

Step2: Use the cosine identity $\cos(-\alpha)=\cos(\alpha)$

Since $\cos(-\alpha)=\cos(\alpha)$, then $\cos(-\frac{7\pi}{12})=\cos(\frac{7\pi}{12})$. And $\frac{7\pi}{12}=\frac{3\pi + 4\pi}{12}=\frac{\pi}{4}+\frac{\pi}{3}$.

Step3: Use the cosine sum formula $\cos(A + B)=\cos A\cos B-\sin A\sin B$

Let $A=\frac{\pi}{4}$ and $B = \frac{\pi}{3}$. We know that $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, $\cos\frac{\pi}{3}=\frac{1}{2}$, and $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$. Then $\cos(\frac{\pi}{4}+\frac{\pi}{3})=\cos\frac{\pi}{4}\cos\frac{\pi}{3}-\sin\frac{\pi}{4}\sin\frac{\pi}{3}=\frac{\sqrt{2}}{2}\times\frac{1}{2}-\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}=\frac{\sqrt{2}-\sqrt{6}}{4}$.

Answer:

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