Properties of Inverse Trigonometric Functions

Introduction

The properties of inverse trigonometric functions are associated with the range as well as domain of the function. Inverse trigonometric functions are identified as the inverse of some basic trigonometric functions such as sine, cosine, tangent, secant, cosecant, and cotangent functions. Inverse trigonometric functions are also known as, arc functions and cyclometric functions. These expressions of inverse trigonometric functions allow you to find any angle at any trigonometric ratio. These expressions are derived from the properties of trigonometric functions.It is expressed as −

$$\mathrm{\sin^{-1}\:,\:\cos^{-1}\:,\:\sec^{-1}\:,\:cosec^{-1}\:,\:\cot^{-1}\:,\:and\:\tan^{-1}}$$

Inverse trigonometric functions also are known as, arc functions, and cyclometric functions. You'll use these inverse trigonometric expressions to find any angle with any trigonometric ratio. In this tutorial, we will discuss the Properties of Inverse Trigonometric Functions.

Trigonometric Functions

Trigonometric functions and identities are the aspect ratios of right triangles. There are six basic trigonometric functions used in trigonometric functions. These functions are trigonometric functions. The six basic trigonometric functions are the sine function, the cosine function, the secant function, the cosecant function, the tangent function, and the cotangent function. The sides of a right triangle are the vertical, hypotenuse, and base used to calculate the values of sine, cosine, tangent, secant, cosecant, and cotangent using its formulas.

Inverse Trigonometric Functions

The inverse of a basic circular function (trigonometric function) is called an inverse trigonometric function. Inverse trigonometric functions are closely related to basic trigonometric functions as a learning topic. In mathematics, an inverse trigonometric function (also known as an arc function, anti-trigonometric function, or cyclometric function) is the inverse of a trigonometric function (the domain of the corresponding definition is limited). The inverse trigonometric function is expressed as − $\mathrm{\sin^{-1}\:,\:\cos^{-1}\:,\:\sec^{-1}\:,\:cosec^{-1}\:,\:\cot^{-1}\:,\:and\:\tan^{-1}}$ The inverse trigonometric function holds all the expressions of a basic trigonometric function.

Properties of Inverse Trigonometric Functions

The basic properties help us to solve the problem. Some important properties are −

• For the inverse of x, the inverse trigonometric function transforms the provided inverse trigonometric function into its inverse. This comes from the cyclic function where sin and the cosecant are the reciprocals, the tangent and the cotangent are the reciprocals, and therefore the cosine and the secant are the reciprocals. Alternatively, inverse trigonometric expressions for inverse sine, inverse cosine, and inverse tangent are often written as −

$$\mathrm{\sin^{-1}\:=\:cosec^{-1}(\frac{1}{x})\:where\:x\:\varepsilon\:[-1\:,\:1]\:-\:\lbrace\:0\:\rbrace}$$

$$\mathrm{\cos^{-1}\:=\:\sec^{-1}(\frac{1}{x})\:where\:x\:\varepsilon\:[-1\:,\:1]\:-\:\lbrace\:0\:\rbrace}$$

$$\mathrm{\tan^{-1}(x)\:=\:\cot^{-1}(\frac{1}{x})\:if\:x>0\:(or)\:\cot^{-1}(\frac{1}{x})\:-\pi\:,\:if\:x<0}$$

$$\mathrm{\cot^{-1}(x)\:=\:\tan^{-1}(\frac{1}{x})\:if\:x>0\:(or)\:\tan^{-1}(\frac{1}{x})\:+\pi\:,\:if\:x<0}$$

• All six trigonometric functions are suitable for arbitrary-valued inverse trigonometric expressions. Negative values are converted to negative values for the sine, tangent, and cosine functions of the inverse trigonometric function. In addition, for cosecant, secant, and cotangent theorem functions, negative values

$$\mathrm{\sin^{-1}(-x)\:=\:-\sin^{-1}(x)}$$

$$\mathrm{\tan^{-1}(-x)\:=\:-\tan^{-1}(x)}$$

$$\mathrm{\cos^{-1}(-x)\:=\:\pi\:-\cos^{-1}(x)}$$

$$\mathrm{\sec^{-1}(-x)\:=\:\pi\:-\:\sec^{-1}(x)}$$

$$\mathrm{\cot^{-1}(-x)\:=\:-cosec^{-1}(x)}$$

$$\mathrm{cosec^{-1}(-x)\:=\:-cosec^{-1}(x)}$$

• $\mathrm{\sin^{-1}(\frac{1}{x})\:=\:cosec^{-1}(x)\:where\:x\geq\:1\:or\:x\leq\:-1}$

$\mathrm{\cos^{-1}(\frac{1}{x})\:=\:\sec^{-1}(x)\:where\:x\geq\:1\:or\:x\leq\:-1}$

$\mathrm{\tan^{-1}(\frac{1}{x})\:=\:-\pi\:+\:\cot^{-1}(x)}$

• $\mathrm{\sin^{-1}(\cos\theta)\:=\:\frac{\pi}{2}\:-\:\theta\:,\:if\:\theta\:\varepsilon\:[0\:,\:\pi]}$

$$\mathrm{\cos^{-1}(\sin\theta)\:=\:\frac{\pi}{2}\:-\:\theta\:,\:\varepsilon\:[-\frac{\pi}{2}\:,\:\frac{\pi}{2}]}$$

$\mathrm{cosec^{-1}(\sec\theta)\:=\:\frac{\pi}{2}\:-\:\theta\:\varepsilon\:[0\:,\:\pi]\:-\:(\frac{\pi}{2})}$

$\mathrm{\sec^{-1}(cosec\theta)\:=\:\frac{\pi}{2}\:-\:\theta\:if\:\theta\:\varepsilon\:[-\frac{\pi}{2}\:,\:0]\:\cup\:[0,\:\frac{\pi}{2}]}$

$\mathrm{\cot^{-1}(\tan\theta)\:=\:\frac{\pi}{2}\:-\:\theta\:if\:\theta\:\varepsilon\:[-\frac{\pi}{2}\:,\:\frac{\pi}{2}]}$

$\mathrm{\tan^{-1}(\cot\theta)\:=\:\frac{\pi}{2}\:-\:\theta\:if\:\theta\:\varepsilon\:[0\:,\:\pi]}$

• Right angles are obtained by adding complementary inverse trigonometric functions. Only the condition is that the x value must be same in both complementary parts that is sum of complementary functions of inverse trigonometry functions will be equal to $\mathrm{\frac{\pi}{2}}$

$$\mathrm{\sin^{-1}(x)\:+\:\cos^{-1}(x)\:=\:\frac{\pi}{2}}$$

$$\mathrm{\tan^{-1}(x)\:+\:\cot^{-1}(x)\:=\:\frac{\pi}{2}}$$

$$\mathrm{\sec^{-1}(x)\:+\:cosec^{-1}(x)\:=\:\frac{\pi}{2}}$$

• The sums and differences of the inverse trigonometric functions were calculated using the given formula. You'll then use these inverse trigonometric expressions to calculate double and triple functional expressions.

$$\mathrm{\sin^{-1}(x)\:+\:\sin^{-1}(y)\:=\:\sin^{-1}[x\:\sqrt{1\:-\:y^{2}}\:+\:y\sqrt{1\:-\:x^{2}}]}$$

$$\mathrm{\cos^{-1}(x)\:+\:\cos^{-1}(y)\:=\:\cos^{-1}[xy\:-\:\sqrt{1\:-\:x^{2}}\sqrt{1\:-\:y^{2}}]}$$

The above formula has been edited from the list of formulas for inverse trigonometric functions. In addition, all basic trigonometric expressions are converted to inverse trigonometric expressions and classified as a set of the above four expressions. Random values, reciprocals and complement functions, sums and differences of functions, double and triple functions.

Domain and Range

In mathematics, an inverse trigonometric function (also known as an arcus function, anti-trigonometric function, or cyclometric function) is the inverse of an inverse trigonometric function (the domain of the corresponding definition is limited). The inverse trigonometric function is expressed as −

$\mathrm{\sin^{-1}\:,\:\cos^{-1}\:,\:\sec^{-1}\:,\:cosec^{-1}\:,\:\cot^{-1}\:,\:and\:\tan^{-1}}$

S.No	Function	Range	Domain
1	$\mathrm{y\:=\:\sin^{-1}(x)}$	$\mathrm{-\frac{\pi}{2}\:\leq\:y\:\leq\:\frac{\pi}{2}}$	$\mathrm{-1\:leq\:x\:\leq\:1}$
2	$\mathrm{y\:=\:\cos^{-1}(x)}$	$\mathrm{0\:\leq\:y\:\leq\:\pi}$	$\mathrm{-1\:leq\:x\:\leq\:1}$
3	$\mathrm{y\:=\:\tan^{-1}(x)}$	$\mathrm{-\frac{\pi}{2}\leq\:y\leq\:\frac{\pi}{2}}$	$\mathrm{x\varepsilon\:R}$
4	$\mathrm{y\:=\:\cot^{-1}(x)}$	$\mathrm{0\:<y\:<\pi}$	$\mathrm{x\:\varepsilon\:R}$
5	$\mathrm{y\:=\:cosec^{-1}(x)}$	$\mathrm{-\frac{\pi}{2}\leq\:y\leq\:\frac{\pi}{2}\:\:y\neq\:0}$	$\mathrm{x\:\leq\:-1\:or\:x\geq\:1}$
6	$\mathrm{y\:=\:\sec^{-1}(x)}$	$\mathrm{0\leq\:y\:\pi\:,\:y\neq\:\frac{\pi}{2}}$	$\mathrm{x\:\leq\:-1\:or\:x\geq\:1}$

Derivative

Inverse trigonometric functions are generally represented by adding an arc to the trigonometric prefix or by adding a power of -1 as follows −

For example. −

We’ll use the example of arcsin. You could use any other inverse function.

• $\mathrm{f(x)\:=\:arc\sin(x)}$

Solve for x, by “undoing” the function on both sides.

$$\mathrm{x\:=\:\sin(f(x))}$$

differentiate both side with respect to x.

$$\mathrm{1\:=\:\cos(f(x))\:f^{'}(x)}$$

Divide to solve for $\mathrm{f^{'}(x)}$

$$\mathrm{\frac{1}{\cos(f(x))}\:=\:f^{'}(x)}$$

Substitute f(x) −

$$\mathrm{\frac{1}{\cos(f(x))}}$$

This is essentially the answer, but some people like to simplify it further with the righttriangle technique:

$$\mathrm{f^{'}(x)\:=\:\frac{1}{\sqrt{1\:-\:y^{2}}}\:=\:\frac{d}{dx}\sin^{-1}(x)}$$

• $\mathrm{f(x)\:=\:arc\tan(x)}$

Solve for x, by “undoing” the function on both sides.

$$\mathrm{x\:=\:\tan(f(x))}$$

Differentiate both side with respect to x

$$\mathrm{1\:=\:\sec^{2}(f(x))f^{'}(x)}$$

$$\mathrm{f(x)\:=\:\frac{1}{\sec^{2}(f(x))}}$$

$$\mathrm{f^{'}(x)\:=\:\frac{1}{\sec^{2}(arc\:tan\:x)}}$$

$$\mathrm{f^{'}(x)\:=\:\frac{1}{1\:+\:x^{2}}\:=\:\frac{d}{dx}\tan^{-1}(x)}$$

The derivative of inverse trigonometric function are follows −

Function	$\mathrm{Derivative\:(\frac{dy}{dx})}$
$\mathrm{y\:=\:\sin^{-1}(x)}$	$\mathrm{\frac{1}{\sqrt{1\:-\:x^{2}}}}$
$\mathrm{y\:=\:\cos^{-1}(x)}$	$\mathrm{-\frac{1}{\sqrt{1\:-\:x^{2}}}}$
$\mathrm{y\:=\:\tan^{-1}(x)}$	$\mathrm{\frac{1}{1\:+\:x^{2}}}$
$\mathrm{y\:=\:\cot^{-1}(x)}$	$\mathrm{-\:\frac{1}{1\:+\:x^{2}}}$
$\mathrm{y\:=\:cosec^{-1}(x)}$	$\mathrm{\frac{1}{\lvert\:x\rvert\:\sqrt{1\:-\:x^{2}}}}$
$\mathrm{y\:=\:\sec^{-1}(x)}$	$\mathrm{-\:\frac{1}{\lvert\:x\rvert\:\sqrt{1\:-\:x^{2}}}}$

Solved Examples

1) Find the derivative of $\mathrm{y\:=\:\tan^{-1}(x)}$

Answer − $\mathrm{y\:=\:\tan^{-1}(x)}$

Solve for x, by “undoing” the function on both sides.

$$\mathrm{x\:=\:\tan(f(x))}$$

Differentiate both sides with respect to x

$$\mathrm{1\:=\:\sec^{2}(f(x))f'(x)}$$

$$\mathrm{f'(x)\:=\:\frac{1}{\sec^{2}(f(x))}}$$

$$\mathrm{f'(x)\:=\:\frac{1}{\sec^{2}(arctanx)}}$$

This is essentially the answer, but some people like to simplify it further with the righttriangle technique −

$$\mathrm{f'(x)\:=\:\frac{1}{1\:+\:x^{2}}\:=\:\frac{d}{dx}\:tan^{-1}(x)}$$

2) Find the derivative of $\mathrm{y\:=\:\sin^{-1}(x)}$

Answer − $\mathrm{y\:=\:\sin^{-1}(x)}$

Solve for x, by “undoing” the function on both sides.

$$\mathrm{x\:=\:\sin(f(x))}$$

Differentiate both sides with respect to x.

$$\mathrm{1\:=\:\cos(f(x))f'(x)}$$

Divide to solve for 𝑓′(𝑥)

$$\mathrm{\frac{1}{\cos(f(x))}\:=\:f'(x)}$$

Substitute f(x) −

$$\mathrm{\frac{1}{\cos(arc\sin(x))}\:=\:f'(x)}$$

This is essentially the answer, but some people like to simplify it further with the righttriangle technique −

$$\mathrm{f'(x)\:=\:\frac{1}{\sqrt{1\:-\:x^{2}}}\:=\:\frac{d}{dx}\sin^{-1}(x)}$$

3) If $\mathrm{\sec^{-1}(2)\:+\:cosec^{-1}(x)\:=\:\frac{\pi}{2}}$ then find the value of x using the property of the inverse trigonometry function

Answer − According to the property of inverse trigonometry function.

$$\mathrm{\sec^{-1}(x)\:+\:cosec^{-1}(x)\:=\:\frac{\pi}{2}}$$

Given equation is $\mathrm{\sec^{-1}(x)\:+\:cosec^{1}(x)\:=\:\frac{\pi}{2}}$

Now on comparing both equations, we get the value x as 2.

4) Find the principal value of $\mathrm{\cos^{-1}(\frac{-1}{2})}$

Answer − Let $\mathrm{y\:=\:\cos^{-1}(\frac{-1}{2})}$

This can be written as −

$\mathrm{\cos\:y\:=\:\frac{-1}{2}}$

$\mathrm{\cos\:y\:=\:\cos(\frac{2\pi}{3})}$

Therefore, the main value range for $\mathrm{\cos^{-1}\:is\:[0\:,\:\pi]}$

Hence, the principal value of $\mathrm{\cos^{-1}(\frac{-1}{2})\:is\:\frac{2\pi}{3}}$

Conclusion

An inverse trigonometric function (also known as an arc function, anti-trigonometric function, or cyclometric function) is the inverse of an inverse trigonometric function (the domain of the corresponding definition is limited). The properties of inverse trigonometric functions are based on the domain and range of the function.

FAQs

1. What do you mean by Inverse trigonometric functions?

The inverse of a basic circular function (trigonometry function) is known as the inverse trigonometric function.

2. What are the six basic trigonometric functions?

These functions are trigonometric functions. The six basic trigonometric functions are the sine function, the cosine function, the secant function, the cosecant function, the tangent function, and the cotangent function

3. What do you mean by trigonometry function?

Trigonometric functions and identities are the aspect ratios of right triangles. There are six basic trigonometric functions used in trigonometric functions sine, cosine, tangent, secant, cosecant, and cotangent. These functions are trigonometric functions.

4. What is the use of inverse trigonometric functions with inverse trigonometric functions?

To find an unknown measure of the angle of a right triangle if you know the length of the two sides

5. What are the range and domain of $\mathrm{\sin^{-1}(x)\:and\:\cos^{-1}(x)}$

Range of $\mathrm{\sin^{-1}(x)\:=\:\frac{-\pi}{2}\:\leq\:y\:\leq\:\frac{\pi}{2}}$

Range of $\mathrm{\cos^{-1}(x)\:=\:0\:\leq\:y\:\leq\:\pi}$

Domain of $\mathrm{\sin^{-1}(x)\:=\:-1\:\leq\:x\:\leq\:1}$

Domain of $\mathrm{\cos^{-1}(x)\:=\:-1\leq\:x\:\leq\:1}$