Question

Find the principal value of ${{\cot }^{-1}}(\cot 4)$ is

Answer 1

Hint: We will first check if 4 radians will lie in the range of ${{\cot }^{-1}}x$ or not. If not then we will find the angle in terms of $\pi$ and which will lie in the range between 0 and $\pi$. Using this information we will get the answer.
Complete step by step solution:
Before proceeding with the question, we should understand the definition of principal value. The principal value of ${{\cot }^{-1}}x$ for x>0, is the length of the arc of a unit circle centred at the origin which subtends an angle at the centre whose cot is x. For this reason it is also denoted by arc cot x.
Now let us assume that ${{\cot }^{-1}}(\cot 4)........(1)$
Also we know that the range is between 0 and $\pi$. So now rearranging the terms in equation (1) we get,
$\Rightarrow {{\cot }^{-1}}(\cot (\pi +(4-\pi ))........(2)$
Now we know that $\cot (\pi +x)=\cot x$ and hence applying in equation (2) we get,$\Rightarrow {{\cot }^{-1}}(\cot (4-\pi ))........(3)$
Now again cancelling and rearranging in equation (3) we get,
$\Rightarrow 4-\pi$
Hence $4-\pi$ is the answer.

Note: We need to be clear about what the principal value in trigonometry means. Also we need to understand the difference between radians and degrees. And we need to remember the range and domains of different inverse trigonometric functions. We in a hurry can make a mistake in applying the cofunction identity $\cot \left( \pi +x \right)=\cot x$ as we can write tan in place of cot in equation (3).