Home » Maths » how to convert tan inverse to sin, cos, sec, cosec x, cot inverse

how to convert tan inverse to sin, cos, sec, cosec x, cot inverse

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

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