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.