what is the exact value of $cos(\frac{11pi}{21})cos(\frac{pi}{7})-sin(\frac{11pi}{21})sin(\frac{pi}{7})$?\n$…

what is the exact value of $cos(\frac{11pi}{21})cos(\frac{pi}{7})-sin(\frac{11pi}{21})sin(\frac{pi}{7})$?\n$-\frac{sqrt{3}}{2}$\n$-\frac{1}{2}$\n$\frac{1}{2}$\n$\frac{sqrt{3}}{2}$

what is the exact value of $cos(\frac{11pi}{21})cos(\frac{pi}{7})-sin(\frac{11pi}{21})sin(\frac{pi}{7})$?\n$-\frac{sqrt{3}}{2}$\n$-\frac{1}{2}$\n$\frac{1}{2}$\n$\frac{sqrt{3}}{2}$

Answer

Explanation:

Step1: Use the cosine addition formula

The formula (\cos(A + B)=\cos A\cos B-\sin A\sin B). Let (A=\frac{11\pi}{21}) and (B = \frac{\pi}{7}=\frac{3\pi}{21}). Then (\cos(\frac{11\pi}{21})\cos(\frac{\pi}{7})-\sin(\frac{11\pi}{21})\sin(\frac{\pi}{7})=\cos(\frac{11\pi}{21}+\frac{3\pi}{21})).

Step2: Simplify the argument of the cosine function

(\frac{11\pi}{21}+\frac{3\pi}{21}=\frac{11\pi + 3\pi}{21}=\frac{14\pi}{21}=\frac{2\pi}{3}).

Step3: Evaluate the cosine function

We know that (\cos(\frac{2\pi}{3})=\cos(120^{\circ})). Using the unit - circle, (\cos(\frac{2\pi}{3})=-\frac{1}{2}).

Answer:

(-\frac{1}{2})