Question

How do I find the value of $\tan \dfrac{\pi}{12}$?

Answer 1

Hint: In the above question we were asked to find the value of$\tan \dfrac{\pi}{12}$. You can solve this problem using the double angle formula. After which you have to use the quadratic formula to get the value of$\tan \dfrac{\pi}{12}$. Also, you have to take care of the quadrants of tan to get the correct value. So let us see how we can solve this problem.

Complete step by step Solution:
In the given question we have to find the value of$\tan \dfrac{\pi}{12}$. We will use the double angle formula of tan to solve this problem.
The formula of the double angle of tan is $\tan 2a = \dfrac{{2\tan a}}{{1 - {{\tan }^2}a}}$
According to our problem,
 $\Rightarrow \tan (\dfrac{\pi }{6}) = \dfrac{{2\tan (\dfrac{\pi }{{12}})}}{{1 - {{\tan }^2}(\dfrac{\pi }{{12}})}} = \dfrac{1}{{\sqrt 3 }}$
After cross-multiplying we get,
 $\Rightarrow 2\sqrt 3 \tan (\dfrac{\pi }{{12}}) = 1 - {\tan ^2}(\dfrac{\pi }{{12}})$
After arranging the above expression we get,
 $\Rightarrow {\tan ^2}(\dfrac{\pi }{{12}}) + 2\sqrt 3 \tan (\dfrac{\pi }{{12}}) - 1 = 0$
From the formula of the quadratic equation for$\tan \dfrac{\pi}{12}$, the discriminant is ${b^2} - 4ac$
For our problem, D = ${d^2} = {b^2} - 4ac$
 $= 12 + 4$
 $= 16$
 $\therefore d = \pm 4$
Now, we know that there are two real roots:
 $\Rightarrow \tan (\dfrac{\pi }{{12}}) = - \dfrac{b}{{2a}} \pm \dfrac{d}{{2a}}$
 $= - \dfrac{{2\sqrt 3 }}{2} \pm \dfrac{4}{2}$
After solving the above expression we get,
 $= - \sqrt 3 \pm 2$

Since $\tan (\pi /12)$ is positive, therefore $\tan (\dfrac{\pi }{{12}}) = 2 - \sqrt 3$.

Note:
In the above solution, we used a quadratic formula. After solving the problem with the quadratic equation we get both positive and negative values. Since tan(pi/12) lies in the first quadrant, therefore it must be positive because every trigonometric identity is positive in the first quadrant.