Inverse Trigonometric Functions

Like the basic concept of inverse functions, the trigonometric inverse function has the same concept related to sine, cosine, tangent, cot, sec, and cosec. As addition is the inverse of subtraction and multiplication is the inverse of division, in the same way, trigonometry inverse functions work opposite to their value. For instance: 

• Sin-1(inverse sine) is the opposite of sine

• Cos-1 (inverse cosine) is the opposite of cosine

• Tan-1 (inverse tangent) is the inverse of a tangent

• Cot-1 (inverse cotangent) works opposite to cot

Determining the value of the angle is possible with the inverse trigonometry function. 

Mathematical Expression of Trig Inverse

Expressing statements mathematically,

Sinθ = Y has the inverse function as sin-1Y = θ.

Similarly,

It works the same for other functions of trigonometry. (-1) is not the exponent here. It simply represents the inverse function. Hence, it does not mean that Sin-1θ is equal to

 1 / Sinθ. Inverse functions in trigonometry have the pretext in their name as –

• Arcsine

• Arccosine

• Arctangent

• Arccotangent

• Arcsecant

• Arccosecant

List of Trigonometry Inverse Functions Formulas Set

Formulas of inverse trigonometric functions discussed below have been grouped to convert one function to another, perform various operations, and determine the principal values of the angles. Besides, formulas of inverse trigonometric functions have been obtained from the basic trig functions formulas. It is classified into the following four sets –

1. Arbitrary values

2. Double and triple of functions

3. Complementary and reciprocal

4. Arithmetic operations of functions

Understanding Trigonometric Inverse Functions for Arbitrary Values

The negative values of inverse functions of sine, cosine, and tangent are transformed into the entire operations’ negative. The negatives of the importance of inverse functions of cosec, sec, and cot result in subtracting the function from the π. 

Double And Triple of Trigonometric Inverse Function Formulas

Both double and triple of the inverse trig functions are solved to get the single trigonometric function. For detailed information, the below table will be useful. 

Reciprocals and Complementary Functions

Inverse trigonometry functions for reciprocal values of x transform the given inverse trig function into a reciprocal function. Reciprocal in trigonometry point of view is as – 

• Sine is the reciprocal of cosecant and vice versa

• Cosine is the reciprocal of secant and vice versa

• Tangent is the reciprocal of cotangent and vice versa

On the other hand, the complementary function is the sum of two functions giving 90 degrees. In terms of trigonometry, the sum of the complementary inverse functions gives the right angle (90 degrees or π/2), as a result. Therefore, the sum of complementary functions such as tangent and cotangent, sine and cosine, secant and cosecant results in π/2. 

Differentiation of Inverse Trigonometric Functions with The Domain and Range

Finding the differentiation of different inverse trig functions is evaluated using some formulas. Besides, the domain and range of trigonometric functions are decided based on graphs of sine inverse, cosine inverse, tangent inverse, cotangent inverse, cosecant inverse, and secant inverse.

Conclusion

It is necessary to memorise the inverse trigonometric functions with their domain and range to solve and evaluate the inverse functions. The concept of reciprocals, double and triple, as well as other arithmetic operations of inverse trigonometric functions, is discussed in brief. We have also mentioned the differentiation formulas for every inverse trigonometry function to ensure easy and quick calculation.

Candidates preparing for entrance exams can check and go through the explanation of inverse trigonometric functions. In simple words, inverse trigonometric functions are the inverse or opposite of the basic functions in trigonometry.