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