Inverse Trigonometric Functions

Inverse trigonometric functions, as a topic of learning, are closely related to the basic trigonometric functions. The domain and the range of the trigonometric functions are converted to the range and domain of the inverse trigonometric functions. In trigonometry, we learn about the relationships between angles and sides in a right-angled triangle. Similarly, we have inverse trigonometry functions. The basic trigonometric functions are sin, cos, tan, cosec, sec, and cot. The inverse trigonometric functions on the other hand are denoted as sin^-1x, cos^-1x, cot^-1 x, tan^-1 x, cosec^-1 x, and sec^-1 x.

Inverse trigonometric functions have all the formulas of the basic trigonometric functions, which include the sum of functions, double and triple of a function. Here we shall try to understand the transformation of the trigonometric formulas to inverse trigonometric formulas.

Inverse trigonometric functions are the inverse functions relating to the basic trigonometric functions. The basic trigonometric function of sin θ = x, can be changed to sin^-1 x = θ. Here, x can have values in whole numbers, decimals, fractions, or exponents. For θ = 30° we have θ = sin^-1 (1/2), where θ lies between 0° to 90°. All the trigonometric formulas can be transformed into inverse trigonometric function formulas.

Inverse trigonometric functions are also known as the anti-trigonometric functions/ arcus functions/ cyclometric functions. Inverse trigonometric functions are the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. The inverse trigonometric functions are written using arc-prefix like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). The inverse trigonometric functions are used to find the angle of a triangle from any of the trigonometric functions. It is used in diverse fields like geometry, engineering, physics, etc.

Consider, the function y = f(x), and x = g(y) then the inverse function can be written as g = f^-1,

This means that if y = f(x), then x = f^-1(y).

An example of inverse trigonometric function is x = sin^-1y.

The list of inverse trigonometric formulas has been grouped under the following formulas. These formulas are helpful to convert one function to another, to find the principal angle values of the functions, and to perform numerous arithmetic operations across these inverse trigonometric functions. Further all the basic trigonometric function formulas have been transformed to the inverse trigonometric function formulas and are classified here as the following four sets of formulas.

• Arbitrary Values
• Reciprocal and Complementary functions
• Sum and difference of functions
• Double and triple of a function

Inverse Trigonometric Function Formulas for Arbitrary Values

The inverse trigonometric function formula for arbitrary values is applicable for all the six trigonometric functions. For the inverse trigonometric functions of sine, tangent, cosecant, the negative of the values are translated as the negatives of the function. And for functions of cosine, secant, cotangent, the negatives of the domain are translated as the subtraction of the function from the π value.

• sin^-1(-x) = -sin^-1x,x ∈ [-1,1]
• tan^-1(-x) = -tan^-1x, x ∈ R
• cosec^-1(-x) = -cosec^-1x, x ∈ R - (-1,1)
• cos^-1(-x) = π - cos^-1x, x ∈ [-1,1]
• sec^-1(-x) = π - sec^-1x, x ∈ R - (-1,1)
• cot^-1(-x) = π - cot^-1x, x ∈ R

Inverse Trigonometric Function Formulas for Reciprocal Functions

The inverse trigonometric function for reciprocal values of x converts the given inverse trigonometric function into its reciprocal function. This follows from the trigonometric functions where sin and cosecant are reciprocal to each other, tangent and cotangent are reciprocal to each other, and cos and secant are reciprocal to each other.

The inverse trigonometric formula of inverse sine, inverse cosine, and inverse tangent can also be expressed in the following forms.

• sin^-1x = cosec^-11/x, x ∈ R - (-1,1)
• cos^-1x = sec^-11/x, x ∈ R - (-1,1)
• tan^-1x = cot^-11/x, x > 0
  tan^-1x = - π + cot^-1 x, x < 0

Inverse Trigonometric Function Formulas for Complementary Functions

The sum of the complementary inverse trigonometric functions results in a right angle. For the same values of x, the sum of complementary inverse trigonometric functions is equal to a right angle. Therefore, complementary functions of sine-cosine, tangent-cotangent, secant-cosecant, sum up to π/2. The complementary functions, sine-cosine, tangent-cotangent, and secant-cosecant can be interpreted as,

• sin^-1x + cos^-1x = π/2, x ∈ [-1,1]
• tan^-1x + cot^-1x = π/2, x ∈ R
• sec^-1x + cosec^-1x = π/2, x ∈ R - [-1,1]

Sum and Difference of Inverse Trigonometric Function Formulas

The sum and the difference of two inverse trigonometric functions can be combined to form a single inverse function, as per the below set of formulas. The sum and the difference of the inverse trigonometric functions have been derived from the trigonometric function formulas of sin(A + B), cos(A + B), tan(A + B). These inverse trigonometric function formulas can be used to further derive the double and triple function formulas.

• sin^-1x + sin^-1y = sin^-1(x.√(1 - y²) + y√(1 - x²))
• sin^-1x - sin^-1y = sin^-1(x.√(1 - y²) - y√(1 - x²))
• cos^-1x + cos^-1y = cos^-1(xy - √(1 - x²).√(1 - y²))
• cos^-1x - cos^-1y = cos^-1(xy + √(1 - x²).√(1 - y²))
• tan^-1x + tan^-1y = tan^-1(x + y)/(1 - xy), if xy < 1
• tan^-1x + tan^-1y = tan^-1(x - y)/(1 + xy), if xy > - 1

Double of Inverse Trigonometric Function Formulas

The double of an inverse trigonometric function can be solved to form a single trigonometric function as per the below set of formulas.

• 2sin^-1x = sin^-1(2x.√(1 - x²))

• 2cos^-1x = cos^-1(2x² - 1)

• 2tan^-1x = tan^-1(2x/1 - x²)

These formulas are derived from the double angle formulas of trigonometry.

Triple of Inverse Trigonometric Function Formulas

The triple of the inverse trigonometric functions can be solved to form a single inverse trigonometric function as per the below set of formulas.

• 3sin^-1x = sin^-1(3x - 4x³)

• 3cos^-1x = cos^-1(4x³ - 3x)

• 3tan^-1x = tan^-1(3x - x³/1 - 3x²)

These formulas resemble and are derived from the triple angle formulas of trigonometry.

Let us study the properties of inverse trigonometric functions using their graph, domain, and range in the following sections.

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

We can plot the graphs of different inverse trigonometric functions with their range of principal values. Here, we have chosen random values for x in the domain of respective inverse trigonometric functions. We will understand the domain and range of these functions in the following sections.

Arcsine Function

Arcsine function or inverse sine function, also denoted as sin^-1 x, is an inverse of the sine function. The graph of sine inverse function is as given below.

Arccosine Function

Arccosine function or inverse cosine function, also denoted as cos^-1 x, is an inverse of the cosine function. The graph of cos inverse function is as given below.

Arctangent Function

Arctangent function or inverse tangent function, also denoted as tan^-1 x, is an inverse of the tan function. The graph of tan inverse function is as given below.

Arccotangent Function

Arccotangent function or inverse cotangent function also denoted as cot^-1 x, is an inverse of the cotangent function. The graph of cot inverse function is as given below.

Arcsecant Function

Arcsecant function or inverse secant function, also denoted as sec^-1 x, is an inverse of the secant function. The graph of sec inverse function is as given below.

Arccosecant Function

Arccosecant function or inverse cosecant function, also denoted as cosec^-1 x, is an inverse of the cosecant function. The graph of the cosec inverse function is as given below.

The above graphs help us to compare and understand the functions y = sin^-1x, y = cos^-1x, y = tan^-1x, y = cot^-1x, y = sec^-1x, and y = cosec^-1x. The domain(x value) of the function is presented along the x-axis and the range(y value) of the inverse trigonometric function is presented along the y-axis. The below-given table gives the domain and range of principal values of inverse trigonometric functions.

We can find the differentiation of different inverse trigonometric functions using differentiation formulas. The following table gives the result of the differentiation of inverse trig functions.

Integrals of Inverse Trig Functions

Here are the integral formulas of inverse trigonometric functions. To see how to derive each one of them, click here.

☛ Related Topics:

• Inverse Trig Functions Calculator
• Trigonometry
• Pythagoras Theorem
• Inverse Operations

Tips and Tricks on Inverse Trigonometric Functions:

Some of the below tips would be helpful in solving and apply the various formulas of inverse trigonometric functions.

• sin^-1(sin x) = x, when -1 ≤ x ≤ 1 (for more information, click here)
• sin(sin^-1x) = x, when -π/2 ≤ x ≤ π/2.
• sin^-1x is different from (sin x)^-1. Also (sin x)^-1 = 1/sinx
• sin^-1x = θ and θ refer to the angle, which is the principal value of this inverse trigonometric function.

How do You Find Inverse Trigonometric Function?

The inverse trigonometric functions of inverse sine, inverse cosine, or inverse tangent can be found from the basic trigonometric ratios. For example, if sin θ = x then θ = sin⁻¹x. In the same way, we can find the other inverse trig functions.

What is the Use of Inverse Trigonometric Formulas in Inverse Trigonometry?

The inverse trigonometry function formula helps to find the angle for the given inverse trigonometric function of the sides of a right-angled triangle (sin⁻¹x = θ)

How Do you Convert the Inverse Trigonometric Function of Sine Inverse to Tan Inverse?

The sine inverse is converted to tan inverse using one of the inverse trigonometric formulas. The conversion formula is as given below.
sin⁻¹ x = tan⁻¹[x/√(1−x²)]

What are Inverse Trigonometric Function Identites for Reciprocal Functions?

The inverse trigonometric function identities for reciprocal functions are given as,

• sin^-1x = cosec^-11/x
• cos^-1x = sec^-11/x
• tan^-1x = cot^-11/x

What are Arcsine, Arccosine, and Arctangent Inverse Trigonometric Functions?

The terms arcsine, arccosine, and arctangent are the inverse ratio of the trigonometric ratios sinθ, cosθ, and tanθ.

• θ = sin^-1x
• θ = cos^-1y
• θ = tan^-1z

What are Inverse Trigonometric Formulas?

Inverse trigonometric functions are the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. Some basic inverse trigonometric formulas are as given below,

• sin^-1(-x) = -sin^-1x
• tan^-1(-x) = -tan^-1x
• cosec^-1(-x) = -cosec^-1x
• cos^-1(-x) = π - cos^-1x
• sec^-1(-x) = π - sec^-1x
• cot^-1(-x) = π - cot^-1x

How do you Solve Inverse Trig Functions?

Inverse trig functions could be solved using the list of inverse trigonometric formulas. Some of these formulas are,

• sin^-1x + sin^-1y = sin^-1(x.√(1 - y²) + y√(1 - x²)), if x and y ≥ 0 and x²+ y² ≤ 1
• sin^-1x - sin^-1y = sin^-1(x.√(1 - y²) - y√(1 - x²)), if x and y ≥ 0 and x²+ y² ≤ 1
• cos^-1x + cos^-1y = cos^-1(xy - √(1 - x²).√(1 - y²)), if x and y ≥ 0 and x²+ y² ≤ 1
• sin^-1x + cos^-1x = π/2
• tan^-1x + cot^-1x = π/2
• sec^-1x + cosec^-1x = π/2

What are the 6 Inverse Trigonometric Functions?

There are six inverse trigonometric functions in trigonometry. These functions are given as, sin^-1x, cos^-1x, cot^-1 x, tan^-1x, cosec^-1x, and sec^-1x.

What is the Range and Domain of Inverse Cosine Trigonometric Function?

The inverse cosine function is written as cos^-1(x) or arccos(x). Inverse functions swap x and y-values, thus the range of inverse cosine is 0 to pi and the domain is -1 to 1.