Question

If $\tan \theta =2$, how do you find the value of ${{\tan }^{3}}\theta $?

Answer 1

Hint: In this problem we have the value of $\tan \theta $ and asked to calculate the value of ${{\tan }^{3}}\theta $. We can say that the value of ${{\tan }^{3}}\theta $ is the $\tan \theta $ times of ${{\tan }^{2}}\theta $ and the value of ${{\tan }^{2}}\theta $ is the $\tan \theta $ times of the $\tan \theta $. So, to calculate the value of ${{\tan }^{3}}\theta $, we will first calculate the value of ${{\tan }^{2}}\theta $. We can calculate the value of ${{\tan }^{2}}\theta $ by multiplying the $\tan \theta $ with the $\tan \theta $ and simplify the obtained value to get the value of ${{\tan }^{2}}\theta $. After getting the value of ${{\tan }^{2}}\theta $, we will again multiply it with $\tan \theta $ to get the value of ${{\tan }^{3}}\theta $. Now we will simplify the obtained equation, then we will get the required result.

Complete step by step answer:
Given that, $\tan \theta =2$.
Now the value of ${{\tan }^{2}}\theta $ can be calculated by multiplying $\tan \theta $ with the same $\tan \theta $. So, multiplying $\tan \theta $ with $\tan \theta $, then we will get
$\Rightarrow \tan \theta \times \tan \theta =2\times 2$
Simplifying the above equation, then we will get
$\Rightarrow {{\tan }^{2}}\theta =4$.
Now we have the value of ${{\tan }^{2}}\theta $ as ${{\tan }^{2}}\theta =4$. For calculating the value of ${{\tan }^{3}}\theta $ we are going to multiply the above ${{\tan }^{2}}\theta $ value with the $\tan \theta $ value, then we will get
$\Rightarrow {{\tan }^{2}}\theta \times \tan \theta =4\times 2$
Simplifying the above equation, then we will get
$\Rightarrow {{\tan }^{3}}\theta =8$.

Hence the value of ${{\tan }^{3}}\theta $ when $\tan \theta =2$ is $8$.

Note: We can also calculate the above value in another method. We can do cubing on the both sides for the value $\tan \theta =2$, then we will get
$\Rightarrow {{\left( \tan \theta \right)}^{3}}={{\left( 2 \right)}^{3}}$
Simplifying the above equation, then we will get
$\Rightarrow {{\tan }^{3}}\theta =8$
From both the methods we got the same result.