Question

What is the exact value of tan 195°?

−1+3√ / 1+3√

−1+3√ / 1−3√

−1−3√ / 1+3√

−1−3√ / 1−3√

Answer 1

\[\tan 195=\tan \left( 180+15 \right)=\tan 15=\tan \left( 45-30 \right)=\frac{ \tan 45-\tan 30 }{ 1+\tan 45\tan 30 }\]
\[=\frac{ 1-\tan 30 }{ 1+\tan 30 }=?\]

Answer 2

Define A=180deg. B=15deg.
\[\tan (A+B) = \frac{ \tan A + \tan B }{ 1 - \tan A tanB }\]
\[\tan A = \tan (180\deg.) = 0\]
So
\[\tan ( A + B ) = \tan B \]

Define B=Θ/2, Θ=30deg.

\[\tan (\frac{ \theta }{ 2 }) = \frac{ 1- \cos \theta }{ \sin }\]
\[= \frac{ 1- \frac{ \sqrt{3} }{ 2 } }{ \frac{ 1 }{ 2 } } = 2 - \sqrt{3}\]

This problem is not like that, isn't it?
\[\frac{ 1 + \sqrt{3} }{ 1 - \sqrt{3} } = \frac{ (1 + \sqrt{3})^{2} }{ 1 - 3 }\]
\[\frac{ 4 + \sqrt{3} }{ -2 } = 2 - \sqrt{3}\]