Question

How do you evaluate ${{\cot }^{-1}}(1)$ ?

Answer 1

Hint: In the given question we were asked to evaluate ${{\cot }^{-1}}(1)$ . We know that the reciprocal of the tangent is cotangent. The value of $\tan (\dfrac{\pi }{4})$ is 1 and it is the same for $\cot (\dfrac{\pi }{4})$. As cot is reciprocal of tan, therefore, $\tan =\dfrac{perpendicular}{base}$ and so, $\cot =\dfrac{base}{perpendicular}$ . So, let us see how we can solve this problem.

Step by step solution:
We have to evaluate ${{\cot }^{-1}}(1)$.
 $\Rightarrow {{\cot }^{-1}}(\dfrac{\pi }{4})$
Therefore, the general value of ${{\cot }^{-1}}(1)$ is $n\pi +\dfrac{\pi }{4}$ , where n is the integer.

The principal value of ${{\cot }^{-1}}(1)$ is $\dfrac{\pi }{4}$.

Additional Information:
We can get the value of tan x by dividing the perpendicular with the base. And cot x is the reciprocal of tan x. So, we can find the value of cot x by dividing the base with perpendicular in a right-angle triangle. The value of $tan{{0}^{\circ }}$ is 0, $tan{{30}^{\circ }}$ is $\dfrac{1}{\sqrt{3}}$ , $tan{{60}^{\circ }}$ is $\sqrt{3}$ and $tan{{90}^{\circ }}$ is undefined.

Note:
We can find tan x when the length of the perpendicular that is the length of the opposite side of the hypotenuse is divided with the hypotenuse, it gives the value of tan x in a right-angle triangle. And we also know that cot x is the reciprocal of tan x. So, cot x can be calculated by dividing the base with the opposite side which is known as perpendicular.