Question

Find the value of tan pi / 24

Answer 1

Answer:

$tan\,\frac{\pi}{24}=-2-\sqrt{3}+2\sqrt{2+\sqrt{3}}$

Step-by-step explanation:

To Find: Value of $tan\,\frac{\pi}{24}$

Formula used to find the value is given by

$tan\,2x=\frac{2tan\,x}{1-tan^2\,x}$

We know that $tan\,\frac{\pi}{6}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$  

let $x=tan\,\frac{\pi}{12}$ and use the formula we get,

$tan\,2\times\frac{\pi}{12}=\frac{2x}{1-x^2}$

$tan\,\frac{\pi}{6}=\frac{2x}{1-x^2}$

$\frac{\sqrt{3}}{3}=\frac{2x}{1-x^2}$

here $x=tan\,\frac{\pi}{12}$

⇒ 6x = √3 -√3x²

⇒ x² + 2√3x -1 = 0  

using quadratic formula we get,

$x=tan\,\frac{\pi}{12}=2-\sqrt{3}$

Now, let $y=tan\,\frac{\pi}{24}$  

$tan\,2\times\frac{\pi}{24}=\frac{2y}{1-y^2}$

$tan\,\frac{\pi}{12}=\frac{2y}{1-y^2}$

$2-\sqrt{3}=\frac{2y}{1-y^2}$  

⇒ ( 2-√3 )y² + 2y - ( 2-√3 ) = 0  

⇒ y² + 2( 2+√3 ) + 1 = 0.  

Again using quadratic formula we get,

$y=tan\,\frac{\pi}{24}=-2-\sqrt{3}+2\sqrt{2+\sqrt{3}}$

Therefore, $tan\,\frac{\pi}{24}=-2-\sqrt{3}+2\sqrt{2+\sqrt{3}}$