We often need to convert inverse of cot to either inverse of cos, sin, tan, sec, cosec x .In this post we will see how to convert cot inverse to sin, cos, tan, sec, cosec x inverse
Inverse of cot to inverse of sin
Case 1
$cot^{-1} x$ and x > 0
Now we can write as
$\theta=cot^{-1} x$
$cot \theta =x$
Now we know that here $\theta \in [0,\pi/2]$, so it is an acute angle
Now it can be written as
$cot \theta =\frac {x}{1} = \frac {base}{perp}$
In Right angle triangle
Now then hyp becomes
$\text{hyp} = \sqrt { 1 +x^2}$
So , $sin \theta = \frac {perp}{hyp} = \frac {1}{\sqrt { 1 +x^2}}$
or
$\theta= sin^{-1} \frac {1}{\sqrt { 1 +x^2}}$
or $cot^{-1} x = sin ^{-1} \frac {1}{\sqrt { 1 +x^2}}$
Case 2
$cot^{-1} x$ and x < 0
So value of the function will be in the range $[\pi/2 , \pi)$
Now we know from the property that
$cot^{-1} (-x)= \pi – cot^{-1} (x)$
This can be written as
$cot^{-1} x = \pi – cot^{-1} |x| =\pi – sin ^{-1} \frac {1}{\sqrt { 1 +x^2}}$
This makes sense also as Range of the cot and sin function differ. We can convert with out worrying about the sign in $[0, \pi/2]$ as it is common
Thus ,we have different formula depending on the values of x
Inverse of cot to inverse of cos
Case 1
$cot^{-1} x$ and x > 0
from the above, we can write that
$cos \theta = \frac {base}{hyp} = \frac {x}{\sqrt {1+x^2}}$
or
$cot^{-1} x = cos^{-1} \frac {x}{\sqrt { 1 +x^2}}$
Case 2
$cot^{-1} x$ and x < 0
So value of the function will be in the range $[\pi/2 , \pi)$
Now we know from the property that
$cot^{-1} (-x)= \pi – cot^{-1} (x)$
$cot^{-1} x = \pi – cot^{-1} |x| = \pi – cos^{-1} \frac {|x|}{\sqrt { 1 +|x|^2}}= cos^{-1} \frac {x}{\sqrt { 1 +x^2}}$
So we have same formula for any values of x
$cot^{-1} x = cos^{-1} \frac {x}{\sqrt { 1 +x^2}}$
Inverse of cot to inverse of tan
Case 1
$cot^{-1} x$ and x > 0
from the above, we can write that
$tan \theta = \frac {perp}{base} = \frac {1}{x}$
or
$cot^{-1} x = tan^{-1} \frac {1}{x}$
Case 2
$cot^{-1} x$ and x < 0
Now we know from the property that
$cot^{-1} (-x)= \pi – cot^{-1} (x)$
$cot^{-1} x = \pi – cot^{-1} |x| = \pi – tan^{-1} \frac {1}{|x|}= \pi + tan^{-1} \frac {1}{x}$
This makes sense also as Range of the cot and tan function differ. We can convert with out worrying about the sign in $[0, \pi/2]$ as it is common
Thus ,we have different formula depending on the values of x
Inverse of cot to inverse of sec
Case 1
$cot^{-1} x$ and x > 0
from the above, we can write that
$sec \theta = \frac {hyp}{base} = \frac {\sqrt {1+x^2}}{x}$
or
$cot^{-1} x = sec^{-1} \frac {\sqrt {1+x^2}}{x}$
Case 2
$cot^{-1} x$ and x < 0
So value of the function will be in the range $[\pi/2 , \pi)$
Now we know from the property that
$cot^{-1} (-x)= \pi – cot^{-1} (x)$
$cot^{-1} x = \pi – cot^{-1} |x| = \pi – sec^{-1} \frac {\sqrt {1+x^2}}{|x|}= sec^{-1} \frac {\sqrt {1+x^2}}{x}$
So we have same formula for any values of x
$cot^{-1} x = sec^{-1} \frac {\sqrt {1+x^2}}{x}$
Inverse of cot to inverse of cosec
Case 1
$cot^{-1} x$ and x > 0
from the above, we can write that
$sec \theta = \frac {hyp}{perp} =\sqrt {1+x^2}$
or
$cot^{-1} x = cosec^{-1} \sqrt {1+x^2}$
Case 2
$cot^{-1} x$ and x < 0
So value of the function will be in the range $[\pi/2 , \pi)$
Now we know from the property that
$cot^{-1} (-x)= \pi – cot^{-1} (x)$
$cot^{-1} x = \pi – cot^{-1} |x| = \pi – cosec^{-1} \sqrt {1+x^2}$
I hope you like this article on how to convert cot inverse to sin, cos, tan, sec, cosec x inverse
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