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

We often need to convert inverse of cosec to either inverse of cos, sin, tan, sec , cot .In this post we will see how we can do it easily

Inverse of cosec to inverse of sin

This is by property

$cosec^{-1} x = sin^{-1} \frac {1}{x}$

Inverse of cosec to inverse of cos

Case 1
$cosec^{-1} x$ and x > 1

Now we can write as

$\theta=cosec^{-1} x$
$cosec \theta =x$

Now we know that here $\theta \in (0,\pi/2]$, so it is an acute angle

Now it can be written as

$cosec \theta =\frac {x}{1} = \frac {hyp}{perp}$

In Right angle triangle

Now then

$base = \sqrt {x^2 -1}$

Now

$cos \theta = \frac {base}{hyp} = \frac {\sqrt {x^2 -1}}{x}$

or $cosec^{-1}x = cos ^{-1} \frac {\sqrt {x^2 -1}}{x}$

Case 1
$cosec^{-1} x$ and x < -1

So value of the function will be in the range $[-\pi/2 , 0)$

Now we know from the property that

$cosec^{-1} (-x) = – cosec^{-1} (x)$

Therefore

$cosec^{-1} (x) = – cosec^{-1} |x| = – cos^{-1} \frac {\sqrt {x^2 -1}}{|x|} =cos^{-1} \frac {\sqrt {x^2 -1}}{x} – \pi $

This makes sense also as Range of the cosec and cos 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 cosec to inverse of tan

Case 1
$cosec^{-1} x$ and x > 1

from the above, we can write that

$tan \theta = \frac {perp}{base} = \frac {1}{\sqrt {x^2-1}}$

or

$cosec^{-1} x = tan^{-1} \frac {1}{\sqrt { x^2 -1}}$

Case 2
$cosec^{-1} x$ and x < -1

Now we know from the property that

$cosec^{-1} (-x) = – cosec^{-1} (x)$

Therefore

$cosec^{-1} (x) = – cosec^{-1} |x| = – tan^{-1} \frac {1}{\sqrt {x^2-1}} $

Thus ,we have different formula depending on the values of x

Inverse of cosec to inverse of sec

Case 1
$cosec^{-1} x$ and x > 1

from the above, we can write that

$sec \theta = \frac {hyp}{base} = \frac {x}{\sqrt {x^2-1}}$

or

$cosec^{-1} x = sec^{-1} \frac {x}{\sqrt { x^2 -1}}$

Case 2
$cosec^{-1} x$ and x < -1

So value of the function will be in the range $[-\pi/2 , 0)$

Now we know from the property that

$cosec^{-1} (-x) = – cosec^{-1} (x)$

Therefore

$cosec^{-1} (x) = – cosec^{-1} |x| = – sec^{-1} \frac {|x|}{\sqrt {x^2 -1}} =sec^{-1} \frac {x}{\sqrt {x^2 -1}} – \pi $

This makes sense also as Range of the cosec and sec 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 cosec to inverse of cot

Case 1
$cosec^{-1} x$ and x > 1

from the above, we can write that

$cot \theta = \frac {base}{perp} = \sqrt {x^2-1}$

or

$cosec^{-1} x = cot^{-1} \sqrt { x^2 -1}$

Case 2
$cosec^{-1} x$ and x < -1

So value of the function will be in the range $[-\pi/2 , 0)$

Now we know from the property that

$cosec^{-1} (-x) = – cosec^{-1} (x)$

Therefore

$cosec^{-1} (x) = – cosec^{-1} |x| = – cot^{-1} \sqrt {x^2 -1} $

This makes sense also as Range of the cosec and cot 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

I hope you like this article on how to convert cosec inverse x to sin, cos, tan, sec, cot inverse

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