Exact value of tan 5pi/12

Question

asked Sep 26, 2022 153k views

1 Answer

Abhishake Gupta · Answer 1 · 2022-09-30T18:12:15+0000

Answer:

$\displaystyle \tan(5\pi)/(12)=2+√(3)$

Explanation:

Tangent Half Angle

Given an angle θ, then:

$\displaystyle \tan {\frac {\theta }{2}}=\frac {\sin \theta }{1+\cos \theta}$

We are required to find:

$\tan(5\pi)/(12)$

But it cannot be found in tables of main angles. We can use the angle

$\theta = (5\pi)/(6)$

And use the formula above to find the required operation. Hence:

$\displaystyle \tan(5\pi)/(12)=\tan((5\pi)/(6))/(2)$

$\displaystyle \tan(5\pi)/(12)=\frac {\sin (5\pi)/(6) }{1+\cos (5\pi)/(6)}$

$\displaystyle \tan(5\pi)/(12)=\frac {(1)/(2) }{1-(√(3))/(2)}$

Operating:

$\displaystyle \tan(5\pi)/(12)=\frac {(1)/(2) }{(2-√(3))/(2)}$

Simplifying:

$\displaystyle \tan(5\pi)/(12)=(1)/(2-√(3))$

Rationalizing:

$\displaystyle \tan(5\pi)/(12)=(1)/(2-√(3))\cdot (2+√(3))/(2+√(3))$

$\displaystyle \tan(5\pi)/(12)=(2+√(3))/(2^2-√(3)^2)$

$\displaystyle \tan(5\pi)/(12)=(2+√(3))/(4-3)$

Finally:

$\boxed{\displaystyle \tan(5\pi)/(12)=2+√(3)}$