Question

How do you find the value of $\tan \left( {\dfrac{\pi }{3}} \right)$?

Answer 1

Hint: In order to solve this question ,calculate it as $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$ taking $\theta \,\,as\,\,\dfrac{\pi }{3}$.

Complete step-by-step answer:
In this we can use the fundamental trigonometric identity that is
$\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
In our question, value of $\theta = \dfrac{\pi }{3}$
Now putting $\theta = \dfrac{\pi }{3}$
$
  \tan \left( {\dfrac{\pi }{3}} \right) = \dfrac{{\sin \left( {\dfrac{\pi }{3}} \right)}}{{\cos \left( {\dfrac{\pi }{3}} \right)}} \\
    \\
 $
From the trigonometric table we know the value of $\sin \left( {\dfrac{\pi }{3}} \right) = \dfrac{{\sqrt 3 }}{2}$and $\cos \left( {\dfrac{\pi }{3}} \right) = \dfrac{1}{2}$
\[
  \tan \left( {\dfrac{\pi }{3}} \right) = \dfrac{{\dfrac{{\sqrt 3 }}{2}}}{{\dfrac{1}{2}}} \\
  \tan \left( {\dfrac{\pi }{3}} \right) = \dfrac{{\sqrt 3 }}{2} \times \dfrac{2}{1} \\
  \tan \left( {\dfrac{\pi }{3}} \right) = \dfrac{{\sqrt 3 }}{{{2}}} \times \dfrac{{{2}}}{1} \\
  \tan \left( {\dfrac{\pi }{3}} \right) = \sqrt 3 \\
    \\
 \]
Therefore ,value of $\tan \left( {\dfrac{\pi }{3}} \right)$ is $\sqrt 3 $

Note: 1. Periodic Function= A function $f(x)$ is said to be a periodic function if there exists a real number T > 0 such that $f(x + T) = f(x)$ for all x.
If T is the smallest positive real number such that $f(x + T) = f(x)$ for all x, then T is called the fundamental period of $f(x)$ .
Since $\sin \,(2n\pi + \theta ) = \sin \theta $ for all values of $\theta $ and n$ \in $N.
2. Even Function – A function $f(x)$ is said to be an even function ,if $f( - x) = f(x)$for all x in its domain.
Odd Function – A function $f(x)$ is said to be an even function ,if $f( - x) = - f(x)$for all x in its domain.
We know that $\sin ( - \theta ) = - \sin \theta .\cos ( - \theta ) = \cos \theta \,and\,\tan ( - \theta ) = - \tan \theta $
Therefore,$\sin \theta $ and $\tan \theta $ and their reciprocals,$\cos ec\theta $ and $\cot \theta $ are odd functions whereas \[\cos \theta \] and its reciprocal \[\sec \theta \] are even functions.
3. Trigonometry is one of the significant branches throughout the entire existence of mathematics and this idea is given by a Greek mathematician Hipparchus.
4.One must be careful while taking values from the trigonometric table and cross-check at least once to avoid any error in the answer.