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

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

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

Inverse of sec to inverse of sin

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

Now we can write as

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

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

Now it can be written as

$sec \theta =\frac {x}{1} = \frac {hyp}{base}$

In Right angle triangle

Now then

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

Now

$sin \theta = \frac {perp}{hyp} = \frac {\sqrt {x^2 -1}}{x}$

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

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

So value of the function will be in the range $(\pi/2 , \pi]$

Now we know from the property that

$sec^{-1} (-x) = \pi – sec^{-1} (x)$

Therefore

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

This makes sense also as Range of the sec 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 sec to inverse of cos

This is by property

$sec^{-1} x = cos^{-1} \frac {1}{x}$

Inverse of sec to inverse of tan

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

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

Case 2

$sec^{-1} x$ and x < -1

Now we know from the property that

$sec^{-1} (-x) = \pi – sec^{-1} (x)$

Therefore

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

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

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

$cosec \theta = \frac {hyp}{perp} = \frac {x}{\sqrt { x^2 -1}}$
so $sec^{-1} x = cosec^{-1} \frac {x}{\sqrt { x^2 -1}}$

Case 2

$sec^{-1} x$ and x < -1

Now we know from the property that

$sec^{-1} (-x) = \pi – sec^{-1} (x)$

Therefore

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

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

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

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

Case 2

$sec^{-1} x$ and x < -1

Now we know from the property that

$sec^{-1} (-x) = \pi – sec^{-1} (x)$

Therefore

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

This makes sense also as Range of the sec 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

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