The Inverse Trigonometric functions are also called as arcus functions, cyclometric functions or anti-trigonometric functions. These functions are used to obtain angle for a given trigonometric value. Inverse trigonometric functions have various application in engineering, geometry, navigation etc.
Representation of functions:
Generally, the inverse trigonometric function are represented by adding arc in prefix for a trigonometric function, or by adding the power of -1, such as:
Inverse of sin x = arcsin(x) or \(\begin{array}{l}\sin^{-1}x\end{array} \)
Let us now find the derivative of Inverse trigonometric function
Example: Find the derivative of a function \(\begin{array}{l}y = \sin^{-1}x\end{array} \)
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Solution:Given \(\begin{array}{l}y = \sin^{-1}x\end{array} \)
…………(i)
\(\begin{array}{l}\Rightarrow x = \sin y\end{array} \)
Differentiating the above equation w.r.t. x, we have:
\(\begin{array}{l}\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}= \frac{1}{\cos y}\end{array} \)
Putting the value of y form (i), we get
\(\begin{array}{l}\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x} = \frac{1}{\cos y} = \frac{1}{\cos (\sin^{-1}x)}\end{array} \)
………..(ii)
From equation (ii), we can see that the value of cos y cannot be equal to 0, as the function would become undefined.
\(\begin{array}{l}\Rightarrow \sin^{-1}x \neq \frac{-\pi}{2}, \frac{\pi}{2}\end{array} \)
i.e. \(\begin{array}{l}x \neq -1,1\end{array} \)
From (i) we have \(\begin{array}{l}y = \sin^{-1}x\end{array} \)
\(\begin{array}{l}\Rightarrow \sin y = \sin (\sin^{-1}x)\end{array} \)
Using property of trigonometric function,
\(\begin{array}{l}\cos^{2}y = 1 – \sin^{2}y = 1 – (\sin (\sin^{-1}x))^{2} = 1 – x^{2}\end{array} \)
\(\begin{array}{l}\Rightarrow \cos y = \sqrt{1 – x^{2}}\end{array} \)
…………(iii)
Now putting the value of (iii) in (ii), we have
\(\begin{array}{l}\frac{\mathrm{d} y}{\mathrm{d} x}= \frac{1}{\sqrt{1-x^{2}}}\end{array} \)
Therefore, the Derivative of Inverse sine function is
\(\begin{array}{l}\frac{\mathrm{d} }{\mathrm{d} x} (\sin^{-1}x)= \frac{1}{\sqrt{1-x^{2}}}\end{array} \)
Derivatives of Inverse trigonometric functions
| Function |
( \(\begin{array}{l}\frac{\mathrm{d} y}{\mathrm{d} x}\end{array} \) ) |
| arcsin x |
\(\begin{array}{l}\frac{1}{\sqrt{1-x^{2}}}\end{array} \) |
| arccos x |
\(\begin{array}{l}-\frac{1}{\sqrt{1-x^{2}}}\end{array} \) |
| arctan x |
\(\begin{array}{l}\frac{1}{1+ x^{2}}\end{array} \) |
| arccot x |
\(\begin{array}{l}-\frac{1}{1+ x^{2}}\end{array} \) |
| arcsec x |
\(\begin{array}{l}\frac{1}{\left | x \right |\sqrt{x^{2}-1}}\end{array} \) |
| arccsc x |
\(\begin{array}{l}-\frac{1}{\left | x \right |\sqrt{x^{2}-1}}\end{array} \) |
| Example:Find the derivative of a function 2 arcsin x – 5 arccsc x.
Solution:Given \(\begin{array}{l}2\; arcsin x – 5\; arccsc x\end{array} \)
\(\begin{array}{l}\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{2}{\sqrt{1-x^{2}}} + \frac{5}{ x \sqrt{x^{2}-1}}\end{array} \)
Further we can factorize the given expression.
Example:Find the derivative of a function \(\begin{array}{l}\sin^{-1} \left ( \frac{1-x^{2}}{1+x^{2}} \right )\end{array} \) .
Solution:Given y = \(\begin{array}{l}\sin^{-1} \left ( \frac{1-x^{2}}{1+x^{2}} \right )\end{array} \)
\(\begin{array}{l}\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{1}{\sqrt{1-\left ( \frac{1-x^{2}}{1+x^{2}} \right )^{2}}} \times \frac{\mathrm{d} }{\mathrm{d} x} \left ( \frac{1-x^{2}}{1+x^{2}} \right )\end{array} \)
\(\begin{array}{l}\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{1}{\sqrt{ \frac{(1+x^{2})^{2}- (1-x^{2})^{2}}{(1+x^{2})^{2}}} } \times \frac{\mathrm{d} }{\mathrm{d} x} \left ( \frac{1-x^{2}}{1+x^{2}} \right )\end{array} \)
\(\begin{array}{l}\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{1+x^{2}}{\sqrt{ (1+x^{4}+2x^{2})- (1+x^{4}-2x^{2})} } \times \left ( \frac{(1+x^{2})(-2x)- (1-x^{2})(2x))}{(1+x^{2})^{2}} \right )\end{array} \)
\(\begin{array}{l}\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{1+x^{2}}{\sqrt{ 4x^{2}} } \times \left ( \frac{(-2x -2x^{3}- 2x + 2x^{3}) }{(1+x^{2})^{2}} \right )\end{array} \)
\(\begin{array}{l}\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{1+x^{2}}{2x} \times \left ( \frac{-4x }{(1+x^{2})^{2}} \right )\end{array} \)
\(\begin{array}{l}\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x} = \frac{-2}{1+x^{2}}\end{array} \)
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Video Lesson on Trigonometry
Nice explain ? thank you so much for all byjus Teachers ??