Inverse Trigonometric Functions

The basic trigonometric functions are sin, cos, tan, cosec, sec, and cot. Inverse trigonometric functions are denoted as sin-1x, cos-1x, tan-1x, cosec-1x, sec-1x, and cot-1x   respectively. The inverse of the trigonometric functions is called inverse trigonometric functions, sin-1x can be substituted for the basic trigonometric function sin = x. x can take the form of whole integers, fractions, decimals, or exponents in this case.

The anti-trigonometric functions, also known as cyclometric functions or arcus functions, are inverse trigonometric functions. To get the angle of a triangle using any of the trigonometric functions, utilise the inverse trigonometric functions sine, cosine, cosecant, secant, cotangent and tangent. It is frequently used in a variety of domains, including geometry, engineering, and physics. arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x), arccot(x), arccot(x), arccot(x), arccot(x), arccot(x), arccot(x), arccot(x), arccot(x), arccot(x), arccot(x), arccot(x), arccot(x) (x). When the remaining side lengths are known, determine the sides of a triangle. 

Consider the function y = f(x), and if x = g(y), the inverse function is g = f -1.

As a result, if y = f(x), then x = f -1(y).

Let f(g(y)) =y and g(f(y)) =x.

Anti-trigonometric functions, arcus functions, and cyclometric functions are all examples of inverse trigonometric functions. The inverse functions of the basic trigonometric functions sine, cosine, cosecant, secant, cotangent, tangent, are known as inverse trigonometric functions.

 The list of inverse trigonometric functions with domain and range value is given below:

 Trigonometric inverse formulas

The given formulas are combined to get a list of inverse trigonometric formulas. These can also be used to change one function into another, calculate angle values and a variety of \ arithmetic operations over inverse trigonometric functions. Further, all the basic functions of trigonometry can be converted into inverse functions formulas. These are divided into groups given below.

• Triple and double of a function

• Reciprocal functions

• Sum and difference of functions

• Arbitrary values

Inverse trigonometric formulas 

sin-1(-x) = -sin-1(x), x ∈ [-1, 1]

cos-1(-x) = π -cos-1(x), x ∈ [-1, 1]

tan-1(-x) = -tan-1(x), x ∈ R

sin-1x + cos-1x = π/2 , x ∈ [-1, 1]

sin-1(1/x) = cosec-1(x), if x ≥ 1 or x ≤ -1

cos-1(1/x) = sec-1(x), if x ≥ 1 or x ≤ -1

2 tan-1 x = sin-1(2x/(1+x2)), |x| ≤ 1

2tan-1 x = cos-1((1-x2)/(1+x2)), x ≥ 0

3sin-1x = sin-1(3x-4x3)

3cos-1x = cos-1(4x3-3x)

sin(sin-1(x)) = x, -1≤ x ≤1

cos(cos-1(x)) = x, -1≤ x ≤1

Properties of inverse trigonometric functions

Inverse trigonometric functions have qualities that are determined by the domain and range of the functions. Certain aspects of inverse trigonometric functions help solve issues and also help better understand this topic. Inverse trigonometric functions are frequently referred to as arc functions.

They create the length of arc required to obtain a given value of a trigonometric function for a given value. The range of values that an inverse function can achieve with the defined domain of the function is defined as the range of values that the inverse function can achieve. A function’s domain can also be defined as the set of independent variables in which the function exists. Inverse trigonometric functions have a fixed range of values.

Sum and difference of inverse trigonometric function formulas

As per the formulas given, the addition and subtraction of two inverse trigonometric functions can together give a single inverse function. The addition and subtraction of these functions are calculated by sin(A + B), cos(A+ B) and tan(A+ B) formulas.

Sin-1(x (1 – y2) + y (1 – x2)) = Sin-1(x (1 – y2) + y(1 – x2))

• Sin-1(x (1 – y2) – y (1 – x2)) = sin-1(x (1 – y2) – y(1 – x2))

• Cos-1(xy – (1 – x2) (1 – y2)) = cos-1(x y – (1 – x2) (1 – y2))

• Cos-1(xy + (1 – x2) (1 – y2)) = cos -1(x y + (1 – x2) (1 – y2))

• If xy is less than one, tan-1 x + tan-1 y = tan-1 (x + y) / (1 – xy).

• If xy > – 1, tan-1x + tan-1y = tan-1(x – y) / (1 + xy).

Inverse trigonometric functions graph

The graphs of several inverse trigonometric functions can be plotted with the range of their principal values. In the domain of corresponding inverse trigonometric functions, we have picked random values for x. In the following sections, we will study the domain and range of these functions.

Arcsine function

The arcsine function, also known as the inverse sine function or sin-¹ x, is the inverse of the sine function.

Arccosine function

The arccosine function, commonly known as the inverse cosine function or cos-¹ x, is the inverse of the cosine function. T

Domain and range of inverse trigonometric functions

The graphs above assist us in comparing and understanding the functions sin-1x, cos-1x, tan-1x, cot-1x, sec-1x, and cosec-1x. The x value is on the x-axis, while the inverse trigonometric function’s range(y value) is plotted on the y-axis.

Conclusion 

This section taught us about inverse trigonometry and various trigonometry functions and properties. Sum and differences of inverse functions were discussed in detail. The important formula and functions are specifically mentioned.