Question

How do you find the exact value of $\tan \dfrac{\pi }{6}$ ?

Answer 1

Hint : In order to find the value of $\tan \dfrac{\pi }{6}$ , we need to simplify it with the trigonometric identities as we know that is $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$ . Substitute the value of $\dfrac{\pi }{6}$ in the formula, get the results for sine and cosine that we know, solve it and we get the value for tan.

Complete step by step solution:
We are given the value of $\tan \dfrac{\pi }{6}$ .
So, according to this all the three trigonometric values of trigonometric identities, we know that:
$\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
Since, we need to find the value for the angle $\dfrac{\pi }{6}$ , substitute it in the above formula and we get:
$\tan \dfrac{\pi }{6} = \dfrac{{\sin \dfrac{\pi }{6}}}{{\cos \dfrac{\pi }{6}}}$ .
From the basic formulas of trigonometry we know that:
$\sin \dfrac{\pi }{6} = \dfrac{1}{2}$ and $\cos \dfrac{\pi }{6} = \dfrac{{\sqrt 3 }}{2}$
Putting these in the above formula and we get:
$\tan \dfrac{\pi }{6} = \dfrac{{\dfrac{1}{2}}}{{\dfrac{{\sqrt 3 }}{2}}}$
And we know that dividing by one number is the same as multiplying by its reciprocal so:
$\tan \dfrac{\pi }{6} = \dfrac{1}{2} \times \dfrac{2}{{\sqrt 3 }}$
Cancelling the $2's$ and rationalising the denominator, we get:
$\tan \dfrac{\pi }{6} = \dfrac{1}{2} \times \dfrac{2}{{\sqrt 3 }} \\\ \tan \dfrac{\pi }{6} = \dfrac{1}{{\sqrt 3 }} \\\ \tan \dfrac{\pi }{6} = \dfrac{1}{{\sqrt 3 }} \times \dfrac{{\sqrt 3 }}{{\sqrt 3 }} = \dfrac{{\sqrt 3 }}{3} \;$
Therefore, the exact value of $\tan \dfrac{\pi }{6}$ is $\dfrac{1}{{\sqrt 3 }}$ or $\dfrac{{\sqrt 3 }}{3}$ or $\approx 0.577$ .
So, the correct answer is “ $\approx 0.577$ ”.

Note : We can also go for the larger method if the formulas are not remembered that is considering a triangle of perpendicular $1$ unit and hypotenuse as $2$ unit with an angle subtended between them is $\dfrac{\pi }{3}$ , find the base value and angle opposite to the perpendicular and solve for sine value, cosine value as we know $\sin \theta = \dfrac{p}{h}$ , etc.
We can also leave the value of tan obtained in the form of fractions rather than converting it into decimal form.