# How do you use half angle formula to find tan 15?

Mar 27, 2018

$\tan \left({15}^{\circ}\right) = \sqrt{3}$

#### Explanation:

Half angle formulae for $\frac{\tan \left(\theta\right)}{2}$

$\textcolor{w h i t e}{\text{XXX")=(sin(theta))/(1+cos(theta))color(white)("xxxxxxxx}}$[1]

$\textcolor{w h i t e}{\text{XXX")=(cos(theta))/(1-sin(theta))color(white)("xxxxxxxx}}$[2]

$\textcolor{w h i t e}{\text{XXX")=+-sqrt(1-cos(theta))/(1+cos(theta))color(white)("xxxx}}$[3]

Since $\sin \left({30}^{\circ}\right) = \frac{1}{2} \textcolor{w h i t e}{\text{xx}}$and$\textcolor{w h i t e}{\text{xx}} \cos \left({30}^{\circ}\right) = \frac{\sqrt{3}}{2}$

We can use [2] (for example) to get
$\tan \left({15}^{\circ}\right) = \tan \left(\frac{{30}^{\circ}}{2}\right) = \frac{\left(\frac{\sqrt{3}}{2}\right)}{\left(1 - \frac{1}{2}\right)} = \sqrt{3}$

Mar 28, 2018

tan 15 = (2 - sqrt3)

#### Explanation:

Use the half angle formula:
$\tan \left(\frac{a}{2}\right) = \frac{1 - \cos a}{\sin} a$
In this case --> $\tan \left(\frac{a}{2}\right) = \tan 15$ --> $\cos a = \cos 30 = \frac{\sqrt{3}}{2}$
--> $\sin a = \sin 30 = \frac{1}{2}$.
The formula becomes:
$\tan 15 = \frac{1 - \frac{\sqrt{3}}{2}}{\frac{1}{2}} = 2 - \sqrt{3}$
Check by calculator.
$2 - \sqrt{3} = 2 - 1.732 = 0.267$
tan 15 = 0.267. Proved.