Question

Which value of $tan^{-1}2$?

Answer 1

Hint: In this question, we have to find the value of the angle when the tan function is equal to 2. Thus, we will apply the tangent inverse series formula to get the solution. First, we will apply the formula of tan inverse x, and then put the value 2 in the place of x. Then, we will apply the basic mathematical rule. After that, we will find the value of tan 75 with the help of trigonometric formulas and make the necessary calculations, to get the solution for the problem.

Complete step-by-step answer:
According to the problem, we have to find the angle of the tan function which is equal to 2.
The trigonometric function given to us is ${{\tan }^{-1}}2$ -------- (1)
Thus, we will apply the tan inverse formula, which is
${{\tan }^{-1}}x=x-\dfrac{{{x}^{3}}}{3}+\dfrac{{{x}^{5}}}{5}-\dfrac{{{x}^{7}}}{7}+...$
So, now we will put x=2 in the above equation, we get
$\Rightarrow {{\tan }^{-1}}2=2-\dfrac{{{2}^{3}}}{3}+\dfrac{{{2}^{5}}}{5}-\dfrac{{{2}^{7}}}{7}+...$
On further solving the above equation, we get
$\Rightarrow {{\tan }^{-1}}2=2-\dfrac{8}{3}+\dfrac{32}{5}-\dfrac{128}{7}+\dfrac{512}{9}-...$
Now, we will write the fractional number into decimal numbers, we get
$\Rightarrow {{\tan }^{-1}}2=2-2.67+6.4-18.29+56.88-...$
Therefore, we get
$\Rightarrow {{\tan }^{-1}}2=\left( 2+6.4+56.88 \right)-\left( 2.67 \right)$ ------- (2)
Thus, we have taken only 2.67 because we know that $\tan 45=1$ , thus the angle foe 2 must be greater than 45 degree and less than 75 degree because,
$\tan 70=\tan \left( 45+30 \right)$
So, applying the trigonometric formula $\tan \left( a+b \right)=\dfrac{\tan a+\tan b}{1-\tan a\tan b}$ in the above equation, we get
$\Rightarrow \tan \left( 45+30 \right)=\dfrac{\tan 45+\tan 30}{1-\tan 45\tan 30}$
Now, applying the value of $\tan 45=1$ and $\tan 30=\dfrac{1}{\sqrt{3}}$ in the above equation, we get
$\Rightarrow \tan \left( 45+30 \right)=\dfrac{1+\dfrac{1}{\sqrt{3}}}{1-1.\dfrac{1}{\sqrt{3}}}$
On further simplification, we get
$\Rightarrow \tan \left( 45+30 \right)=\dfrac{\dfrac{\sqrt{3}+1}{\sqrt{3}}}{\dfrac{\sqrt{3}-1}{\sqrt{3}}}$
Now, we will put the value off square root of 3 in the above equation, we get
$\Rightarrow \tan \left( 45+30 \right)=\dfrac{1.73+1}{1.73-1}$
Therefore, we get
$\Rightarrow \tan \left( 45+30 \right)=\dfrac{2.73}{0.73}$
$\Rightarrow \tan \left( 45+30 \right)=3.73$
Thus, the value of tan75 is greater than 2, thus we will decrease the angle through 2.67.
Also, we have neglected 18.29 in equation (2) because if we decrease this value, then the angle will become smaller and the value will become less than 2. Thus, equation (2) will become
$\Rightarrow {{\tan }^{-1}}2=65.28-2.67$
$\Rightarrow {{\tan }^{-1}}2=62.61$ which is approximately equal to 63.
Thus, the value of ${{\tan }^{-1}}2$ is equal to 63.43.

Note: While solving this problem, do the step-by-step calculations to avoid error and confusion. Always remember that the formula for tan inverse function has one (+) sign and then (-) and then again (+) sign and so on. Also, remember the formula of tan 75, to get the required solution for the problem.