Question

How do you calculate ${\cot ^{ - 1}}0.684?$

Answer 1

Hint : Make sure what ${\cot ^{ - 1}}0.684$ actually means, if $- 1$ is indicating the power of the given function or the inverse function.
If it is in the power of the function then just change $\cot$ to $\tan$ and find the required value.
But if, $- 1$ is showing the inverse function then use inverse trigonometry in order to solve this problem.

Complete step-by-step answer :
The solution for given question have two possibilities-
If ${\cot ^{ - 1}}0.684$ means the power of $\csc 0.5$ is $- 1$
And if ${\csc ^{ - 1}}0.5$ it is an inverse trigonometric function.
So let us go with the first possibility, that is $- 1$ in ${\csc ^{ - 1}}0.5$ is indicating the power of $\csc 0.684$
Power of a function $- 1$ means the following:
${f^{ - 1}}(x) = \dfrac{1}{{f(x)}}$
So doing this with given function we will get
${\cot ^{ - 1}}0.684 = \dfrac{1}{{\cot 0.684}}$
Now we all know that $\tan x = \dfrac{1}{{\cot x}}$ , so substituting it above, we will get
$\dfrac{1}{{\cot 0.684}} = \tan 0.684$
Here find the value of $\tan 0.684$ in your calculator you will get a value close to $0.8153..$
Now solving the second possibility, that is ${\cot ^{ - 1}}0.684$ is an inverse trigonometric function
Let us assume, ${\cot ^{ - 1}}0.684 = x$ then
We can write $0.684 = \cot x$
Now we know that $\tan x = \dfrac{1}{{\cot x}}$ therefore we can write it as
$0.684 = \dfrac{1}{{\tan x}} \\\ \Rightarrow \tan x = 1.462 \\\ \Rightarrow x = {\tan ^{ - 1}}(1.462) \;$
We have to calculate this value using a calculator, to calculate the inverse function press shift key on your calculator and then press $\tan$ and put the value, the answer will come close to $0.970893251$ in radians and ${55.6^ \circ }$ in degrees.
So, the correct answer is “ ${55.6^ \circ }$ ”.

Note : Generally inverse trigonometric functions have prefix arc written before them to indicate that it is an inverse trigonometric function, e.g. $\arcsin ,\;\arccos$
In this question the argument of $\cot$ is written in radians, so keep this in mind when doing the calculations.