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

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

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

Inverse of sin to inverse of cos

Case 1
$sin^{-1} x$ and x > 0

Now we can write as

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

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

Now it can be written as

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

In Right angle triangle

Now then base becomes
$\text{Base} = \sqrt { 1 -x^2}$

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

or

$\theta= cos^{-1} \sqrt { 1 -x^2}$
or $sin^{-1} x = cos^{-1} \sqrt { 1 -x^2}$

Case II
$sin^{-1} x$ and x < 0

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

Now we know from the property that

$sin^{-1} (-x)= – sin^{-1} (x)$

This can be written as
$sin^{-1} x = – sin^{-1} |x| = – cos^{-1} \sqrt { 1 -|x|^2}= – cos^{-1} \sqrt { 1 -x^2}$

This makes sense also as Range of the cos and sin function differ. We can convert with out worrying about the sign in $[0, \pi/2] as it is common Thus ,we have different formula depending on the values of x Inverse of sin to inverse of tan Case 1$sin^{-1} x$and x > 0 from the above, we can write that$tan \theta = \frac {perp}{base} = \frac {x}{\sqrt {1-x^2}}$or$sin^{-1} x = tan^{-1} \frac {x}{\sqrt { 1 -x^2}}$Case II$sin^{-1} x$and x < 0 Now we know from the property that$sin^{-1} (-x)= – sin^{-1} (x)$This can be written as$sin^{-1} x = – sin^{-1} |x| = – tan^{-1} \frac {x}{\sqrt { 1 -x^2}}= tan^{-1} \frac {-|x|}{\sqrt { 1 -x^2}} = tan^{-1} \frac {x}{\sqrt { 1 -x^2}}$So we have same formula for any values of x$sin^{-1} x = tan^{-1} \frac {x}{\sqrt { 1 -x^2}}$Inverse of sin to inverse of cosec This we already know from the property$sin^{-1} x = cosec^{-1} \frac {1}{x}$for all values of x [-1,1] Inverse of sin to inverse of sec Case 1$sin^{-1} x$and x > 0 from the above, we can write that$sec \theta = \frac {hyp}{base} = \frac {1}{\sqrt {1-x^2}}$or$sin^{-1} x = sec^{-1} \frac {1}{\sqrt { 1 -x^2}}$Case II$sin^{-1} x$and x < 0 Now we know from the property that$sin^{-1} (-x)= – sin^{-1} (x)$This can be written as$sin^{-1} x = – sin^{-1} |x| = – sec^{-1} \frac {1}{\sqrt { 1 -x^2}} = – sec^{-1} \frac {1}{\sqrt { 1 -x^2}}$This makes sense also as Range of the sec and sin function differ. We can convert with out worrying about the sign in$[0, \pi/2] as it is common

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

Inverse of sin to inverse of cot

Case 1
$sin^{-1} x$ and x > 0

from the above, we can write that

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

or

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

Case II
$sin^{-1} x$ and x < 0

Now we know from the property that

$sin^{-1} (-x)= – sin^{-1} (x)$

This can be written as
$sin^{-1} x = – sin^{-1} |x| = – cot^{-1} \frac {\sqrt { 1 -x^2}}{|x|}$

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

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