Chapter 14, Problem 36CT

Solution

To Find:The value of $\tan 60°$ using double angle identity.

The value of the expression is $\tan {60}^0=\sqrt{3}$ .

Given information:

The trigonimetricexpression $\tan 60°$

Concept and Formula Used:

Use the trigonometric identity $\tan 2A=\frac{2\tan A}{1-{\tan }^{2}A}$ .

Calculation:

The given expression is $\tan 60°$

This can be written as:

  $\tan 30°=\tan (2\cdot 30)°$

Using the identity $\tan 2A=\frac{2\tan A}{1-{\tan }^{2}A}$ and simplify:

  $\begin{array}{l}\tan {60}^0=\frac{2\tan {30}^0}{1-{\tan }^2{30}^0}\\ \tan 60°=\frac{2\frac{1}{\sqrt{3}}}{1-{\left(\frac{1}{\sqrt{3}}\right)}^2}\left(\because \tan {30}^0=\frac{1}{\sqrt{3}}\right)\\ \tan 60°=\frac{\frac{2}{\sqrt{3}}}{1-\frac{1}{3}}\\ \tan {60}^0=\frac{\frac{2}{\sqrt{3}}}{\frac{2}{3}}\\ \tan {60}^0=\sqrt{3}\end{array}$

Conclusion:

The value of the expression is $\tan {60}^0=\sqrt{3}$ .