Question

How do you find the exact values of ${{\tan }^{-1}}1$ ?

Answer 1

Hint: To find the exact values of ${{\tan }^{-1}}1$, we are going to first of all equate this inverse expression to $\theta $ then we are going to take tan on both the sides of the equation. On doing that, you will need to use the property which says that multiplying a term with its inverse will give us the answer as 1. Then we should know the angle when $\tan \theta $ takes the value 1.

Complete step-by-step answer:
In the above problem, we are asked to find the exact values of ${{\tan }^{-1}}1$. To find the exact values of ${{\tan }^{-1}}1$, we are going to equate this inverse expression to $\theta $. So, equating ${{\tan }^{-1}}1$ to $\theta $ we get,
${{\tan }^{-1}}1=\theta $
In the above expression, the exact values are the values that $\theta $ can take so for finding the values of $\theta $ we are going to take tan on both the sides of the above equation and we get,
$\tan {{\tan }^{-1}}1=\tan \theta $
Now, we know the property that when a term and its inverse gets multiplied then the result of this multiplication is 1 so the result of the expression $\tan {{\tan }^{-1}}$ is equal to 1 so substituting the value 1 in place of $\tan {{\tan }^{-1}}$ in the above equation we get,
$1=\tan \theta $
Now, we know that the $\theta $ where $\tan \theta $ will take value 1 is $\dfrac{\pi }{4}$ so the value of $\theta $ is equal to $\dfrac{\pi }{4}$.
Also, we know that tan is positive in third quadrant so another angle which will be possible is as follows:
$\pi +\dfrac{\pi }{4}$
Solving the above addition we get,
$\dfrac{4\pi +\pi }{4}=\dfrac{5\pi }{4}$
Hence, the two exact values which we are getting are $\dfrac{\pi }{4},\dfrac{5\pi }{4}$.

Note: The mistake that could be possible in the above solution is that you might forget to write the other angle which lies in the third quadrant so make sure you have put this third quadrant angle also in the final solution.