Question

Exact value of tan 5pi/12

Answer 1

Answer:

$\displaystyle \tan\frac{5\pi}{12}=2+\sqrt{3}$

Step-by-step 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:

$\tan\frac{5\pi}{12}$

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

$\theta = \frac{5\pi}{6}$

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

$\displaystyle \tan\frac{5\pi}{12}=\tan\frac{\frac{5\pi}{6}}{2}$

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

$\displaystyle \tan\frac{5\pi}{12}=\frac {\frac{1}{2} }{1-\frac{\sqrt{3}}{2}}$

Operating:

$\displaystyle \tan\frac{5\pi}{12}=\frac {\frac{1}{2} }{\frac{2-\sqrt{3}}{2}}$

Simplifying:

$\displaystyle \tan\frac{5\pi}{12}=\frac{1}{2-\sqrt{3}}$

Rationalizing:

$\displaystyle \tan\frac{5\pi}{12}=\frac{1}{2-\sqrt{3}}\cdot \frac{2+\sqrt{3}}{2+\sqrt{3}}$

$\displaystyle \tan\frac{5\pi}{12}=\frac{2+\sqrt{3}}{2^2-\sqrt{3}^2}$

$\displaystyle \tan\frac{5\pi}{12}=\frac{2+\sqrt{3}}{4-3}$

Finally:

$\boxed{\displaystyle \tan\frac{5\pi}{12}=2+\sqrt{3}}$