Question

find exact value of tan165

Answer 1

tan(165) = tan(120 + 45) = ?

Answer 2

\[
\tan(165) = \tan(120 + 45) = \frac{\tan(120)+\tan(45)}{1 - \tan(120)\tan(45)} \\
\tan(45) = 1 \\
\tan(60) = \sqrt{3} \\
\text{tan is negative in the second quadrant. Therefore, } \tan(120) = -\sqrt{3} \\
\tan(165) = \frac{-\sqrt{3}+1}{1 +\sqrt{3}} =
\frac{-\sqrt{3}+1}{1 +\sqrt{3}} * \frac{1-\sqrt{3}}{1 -\sqrt{3}} = \\
\frac{1-2\sqrt{3}+3}{1 - 3} = \frac{4-2\sqrt{3}}{-2} =
-2 + \sqrt{3}
\]

Answer 3

THANK YOU SO MUCH