Question

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

Answer 1

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.