Inverse trigonometric functions as the name suggests are the inverse functions of the basic trigonometric functions. Every mathematical function, from the easiest to the most complex, holds an inverse, or opposite function. In addition, the inverse is subtraction similarly for multiplication; the inverse is division. And for trigonometric functions there exist inverse trigonometric functions.

The inverse trigonometry functions are written by applying arc-prefix to the basic functions like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x)and arccot(x). With this article on inverse trigonometric functions learn more about inverse trig functions with tables, formulas, graphs and solved questions.

What are Inverse Trigonometric Functions?

The different names given to the inverse trigonometric functions are anti-trigonometric functions or arcus functions/arc functions or cyclometric functions as for a given value of trigonometric functions, they provide the length of arc required to obtain that particular value.

The inverse trig functions include all the formulas of the basic trigonometric functions, that is the sum of functions, complementary inverse trigonometric functions, double and triple angle of a function and so on.

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For example:

The inverse of p is denoted by \(p^{-1}\).

Let p = f(p) = sin x, then its inverse is \(p=\sin^{-1} x.\)

The inverse sin\( \left(\sin^{-1}x\right)\) does the opposite of the sin. Inverse cosine\( \left(\cos^{-1}x\right) \)does the opposite of cosine and so for the other functions.

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Check out the below table for all the notation of inverse functions:

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Inverse Trigonometric Formulas

The list of various inverse trigonometric functions formulas tables is listed below. These formulas are essential and would be helpful to transform one function to another, to determine the principal angle values of the functions, and to solve numerous arithmetic operations across these inverse trigonometric functions. So let us check them out one by one:

Inverse Trigonometric Function Formula for Reciprocal Values

The inverse trigonometric function for reciprocal values of x transforms the given inverse trigonometric function into its corresponding reciprocal function. This matches the trigonometric functions wherein sin and cosec are reciprocal of one another similarly tan and cot are reciprocal to each other, and cos and sec are reciprocal to each other.

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Inverse Trigonometric Function Formula for Arbitrary Values

The inverse trigonometric functions are applied to determine the angle of a triangle from any of the trigonometric functions and are hence employed in various fields like geometry, engineering design, physics, etc. The inverse trigonometric function formula for arbitrary values for all the six trigonometric functions are listed below.

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Complementary Inverse Trigonometric Functions

The sum of the complementary inverse trigonometric functions appears at a right angle. For the identical values of x, the sum of complementary inverse trigonometric functions is equivalent to a right angle. Hence, complementary functions of sine-cos, tangent-cotangent, sec-cosec, sum up to π/2. The formula interpretation for these are as shown:

Double of Inverse Trigonometric Functions

Trigonometric functions are the functions of angles and sides. The term ”function” is related to describing the association between two sets of numbers/variables. The double of an inverse trigonometric function can be cracked to construct a single trigonometric function with the help of the below-listed formulas:

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Triple of Inverse Trigonometric Functions

In mathematics, there are six fundamental trigonometric functions: sine, cosine, tangent, secant, cosecant, and cotangent. The triple of the inverse trigonometric functions can be translated to construct a single inverse trigonometric function with the help of the formulas listed below:

Sum and Difference of Inverse Trigonometric Function Formulas

The sum or sum and the difference of two inverse trigonometric functions can be merged to form a single inverse function by applying the below formulas:

Domain and Range of Inverse Trigonometric Functions

The trigonometric functions are not one-one and onto their natural domain and range. So, to define inverse trigonometric functions we have to make them one-one and onto functions by restricting their domain and codomain. The inverse trigonometric table given below gives the domain and range of the trigonometric functions.

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Inverse Trigonometric Functions Derivatives

We can locate the differentiation of various inverse trigonometric functions using the formulas listed below.

Inverse Trigonometric Functions Graphs

We can plot the graphs of different inverse trigonometric functions with their range to be the primary branch. Here, we have picked random values for x in the domain of particular inverse trigonometric functions. Along with the graphs, we will understand the domain and range of these functions in this section.

The inverse of six primary trigonometric functions are as follows:

• Arcsine
• Arccosine
• Arctangent
• Arccotangent
• Arcsecant
• Arccosecant

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Arcsine function or the inverse sine function also is denoted as \(\sin^{-1}x\), is the inverse of the sine function. The graph of the sine function along with the inverse function is as shown below.

Sin x graph:

Arcsine graph:

Arccosine/arccos function is the inverse of the cosine function and is denoted as \(\cos^{-1}x\). The graph of the cos function along with the inverse function is as shown below.

Cos x graph:

Arccosine graph:

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Sec x graph:

Arcsecant/arcsec function or inverse secant function is denoted as \(\sec^{-1}x\), which is the inverse of the sec function. The graph of the sec function along with the inverse function is as shown below.

Arcsecant graph:

Arccosecant/arccsc function or inverse cosecant function is denoted as \(\operatorname{cosec}^{-1}x\), which is the inverse of the cosec function. The graph of the cosec function along with the inverse function is as shown below.

Cosec x graph:

Arccosecant graph:

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Tan x graph:

Arctangent/arctan function or the inverse tangent function is denoted as \(\tan^{-1}x\), which is an inverse of the tan function. The graph of the tan function along with the inverse function is as shown below.

Arctangent graph:

cot x graph:

Arccotangent/arccot function or inverse tangent function is denoted as \(\cot^{-1}x\), which is the inverse of the cot function. The graph of the cot function along with the inverse function is as shown below.

Arccotangent graph:

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Solved Examples on Inverse Trigonometric Functions

Check out some of the inverse trig functions questions for more understanding:

Question1:- What is \(\tan^{-1}\left(\frac{1}{4}\right)+\tan^{-1}\left(\frac{3}{5}\right) \)equal to?

Solution:

Concept:

Inverse trigonometry formulas:

\(\tan^{-1}A+\tan^{-1}B=\tan^{-1}\left(\frac{A+B}{1-AB}\right) \)where A > 0, B > 0 & AB < 1

Calculation:

We know that, \(\tan^{-1}A+\tan^{-1}B=\tan^{-1}\left(\frac{A+B}{1-AB}\right)\) where A > 0, B > 0 & AB < 1

\(\Rightarrow\tan^{-1}\left(\frac{1}{4}\right)+\tan^{-1}\left(\frac{3}{5}\right)=\tan^{-1}\left(\frac{\frac{1}{4}+\frac{3}{5}}{1-\frac{1}{4}\times\frac{3}{5}}\right)\)

\(\Rightarrow\tan^{-1}\left(\frac{\frac{17}{20}}{1-\frac{3}{20}}\right)\)

\(\Rightarrow\tan^{-1}\left(1\right)\)

\(\Rightarrow\tan^{-1}\left(\tan\frac{\pi}{4}\right)\)

\(\Rightarrow\frac{\pi}{4}\)

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Question 2:-. What is the value of \(\cos\left[2\sin^{-1}\left(\frac{3}{4}\right)+\cos^{-1}\left(\frac{3}{4}\right)\right]\)?

Solution:

Calculation:

Using Inverse Trigonometry,

\(2\sin^{-1}\left(\frac{3}{4}\right)+\cos^{-1}\left(\frac{3}{4}\right)\)

=\(\sin^{-1}\left(\frac{3}{4}\right)+\sin^{-1}\left(\frac{3}{4}\right)+\cos^{-1}\left(\frac{3}{4}\right)\)

=\(\sin^{-1}\left(\frac{3}{4}\right)+\frac{\pi}{2}\)

Now,

\(\cos\left[2\sin^{-1}\left(\frac{3}{4}\right)+\cos^{-1}\left(\frac{3}{4}\right)\right]=\cos\left[\sin^{-1}\left(\frac{3}{4}\right)+\frac{\pi}{2}\right]\) (By using Trigonometry.)

\(-\sin\left(\sin^{-1}\left(\frac{3}{4}\right)\right)\)

And finally by using Inverse Trigonometry again,-\(\sin\left(\sin^{-1}\left(\frac{3}{4}\right)\right)=\frac{-3}{4}.\)

We hope that the above article on Inverse Trigonometric Functions is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.