Question

Find the exact value. Do not use a calculator.

$\cos \left(7\mathrm{\pi }\right)$

Short answer

The exact value of$\cos \left(7\mathrm{\pi }\right)=-1$

Step-by-step solution

Step 1. Given information

$\cos \left(7\mathrm{\pi }\right)$

Step 2. Unit circle

$$\begin{gathered} \mathrm{Rotate}\mathrm{anticlockwise}\mathrm{to}\mathrm{form}7\mathrm{\pi }\mathrm{angle}\mathrm{with}\mathrm{the}\mathrm{positive}\mathrm{x}-\mathrm{axis}. \\ \mathrm{The}\cos \left(7\mathrm{\pi }\right)\mathrm{equals}\mathrm{the}\mathrm{x}-\mathrm{coordinate}(-1)\mathrm{of}\mathrm{the}\mathrm{point}\mathrm{of}\mathrm{intersection}(-1,0)\mathrm{of}\mathrm{unit}\mathrm{circle}. \end{gathered}$$

Step 3. Final answer

$\cos \left(7\mathrm{\pi }\right)=\mathrm{x}=-1$