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