Question

How do you calculate the inverse cotangent of -2?

Answer 1

Hint: The inverse trigonometric functions give the value of an angle that lies in their respective principal range. The principal range for all inverse trigonometric functions is different.
For \[{{\sin }^{-1}}x\] it is \[\left[ \dfrac{-\pi }{2},\dfrac{\pi }{2} \right]\], for \[{{\cos }^{-1}}x\] it is \[\left[ 0,\pi \right]\], for \[{{\tan }^{-1}}x\] it is \[\left( \dfrac{-\pi }{2},\dfrac{\pi }{2} \right)\], for \[{{\cot }^{-1}}x\] it is \[\left( 0,\pi \right)\], for \[{{\sec }^{-1}}x\] it is \[\left[ 0,\dfrac{\pi }{2} \right)\bigcup \left( \dfrac{\pi }{2},\pi \right]\], and for \[{{\csc }^{-1}}x\] it is \[\left[ -\dfrac{\pi }{2},0 \right)\bigcup \left( 0,\dfrac{\pi }{2} \right]\].

Complete step by step answer:
We are asked to find the inverse cotangent of -2, which means we have to find the value of the \[{{\cot }^{-1}}\left( -2 \right)\]. We know that the inverse trigonometric functions \[{{T}^{-1}}\left( x \right)\], where \[T\] is a trigonometric function gives the value of an angle that lies in their respective principal range. The principal range for the inverse trigonometric function \[{{\cot }^{-1}}\left( x \right)\] is \[\left( 0,\pi \right)\].
We have to find the value of \[{{\cot }^{-1}}\left( -2 \right)\], which means here \[x=-2\]. Let’s assume the value of \[{{\cot }^{-1}}\left( -2 \right)\] is \[y\]. \[y\] is an angle in the principal range of \[{{\cot }^{-1}}\left( x \right)\].
Hence, \[{{\cot }^{-1}}\left( -2 \right)=y\]
Taking \[\cot \] of both sides of the above equation we get,
\[\Rightarrow \cot \left( {{\cot }^{-1}}\left( -2 \right) \right)=\cot \left( y \right)\]
We know that \[T\left( {{T}^{-1}}\left( x \right) \right)=x\], \[T\] is an inverse trigonometric function. Using this property in the above equation we get,
\[\Rightarrow \cot \left( y \right)=-2\]
As we know y is an angle in the range of \[\left( 0,\pi \right)\]. Whose cotangent gives \[-2\]. In the range of \[\left( 0,\pi \right)\] there is only one such angle whose cotangent gives \[-2\]. It is \[2.67795\] rad. Hence \[y=2.67795\] rad.
So, the value of \[{{\cot }^{-1}}\left( -2 \right)\] is \[2.67795\] rad.

Note:
 Generally inverse trigonometric functions will be asked for only those values, for which the angle can be found easily. The principal range of all inverse trigonometric functions should be remembered.