# How do you use the sum and difference identities to find the exact value of tan 105 degrees?

Oct 24, 2014

The special triangles, $30 : 60 : 90 \mathmr{and} 45 : 45 : 90$, allow us to evaluate sine and cosine and tangent.

We leverage that information to evaluate tan(105).

$\tan \left(a + b\right) = \frac{\tan \left(a\right) + \tan \left(b\right)}{1 - \tan \left(a\right) \tan \left(b\right)}$

$\tan \left(60 + 45\right) = \frac{\tan \left(60\right) + \tan \left(45\right)}{1 - \tan \left(60\right) \tan \left(45\right)}$

$\tan \left(105\right) = \frac{\tan \left(60\right) + \tan \left(45\right)}{1 - \tan \left(60\right) \tan \left(45\right)}$

$\tan \left(105\right) = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1}$

$\tan \left(105\right) = \frac{\sqrt{3} + 1}{1 - \sqrt{3}}$

Rationalize

$\tan \left(105\right) = \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \cdot \frac{1 + \sqrt{3}}{1 + \sqrt{3}}$

$\tan \left(105\right) = \frac{\sqrt{3} + 3 + 1 + \sqrt{3}}{1 - 3}$

$\tan \left(105\right) = \frac{2 \sqrt{3} + 4}{- 2}$

$\tan \left(105\right) = - 3.732050808$

To evaluate make sure that the calculator is in Degree mode.