Inverse Function – Definition, Formula, Graph, Examples

Imagine you’re walking down a bridge on a scorching day, and suddenly the thought of measuring the distance between the sun and ground flashes through your mind. Will you be able to calculate the elevation? Or measure the sun’s angle that it is making with the ground on that bridge? That’s where inverse trigonometric functions come in handy. 

Unlike civil engineers, you don’t have to be a master in trigonometry. Understanding the basics of inverse trigonometry will help you measure roof inclinations, slopes, the height and width of a building, light angles and sun shading, installing ceramic tiles and stones, and many more things that a civil engineer does. 

Isn’t it getting interesting? Plus, if you understand inverse functions, it will be easier for you to score more in the examinations. Now, coming to what does the term inverse functions mean? 

What are inverse trigonometric functions? 

Inverse trigonometric functions are inverse functions of the fundamental trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. Arcus, anti-trigonometric, and cyclomatic are other names for these functions. The angle may be calculated using trigonometry ratios using these trigonometry inverse functions. Inverse trigonometry functions are commonly used in engineering, physics, geometry, and navigation.

Inverse Trigonometric Formulas

After understanding what an inverse function means, let’s check some crucial formulas. It is important to understand the limits of the functions as the value might change in accordance with the inverse function. The most common trigonometric formulas are as follows:

Inverse trigonometric functions	Formulas
Arcsine	sin-1(-x) = -sin-1 (x), x ∈ [-1, 1]
Arccosine	cos-1(-x) = π -cos-1 (x), x ∈ [-1, 1]
Arctangent	tan-1(-x) = -tan-1 (x), x ∈ R
Arccotangent	cot-1(-x) = π – cot-1 (x), x ∈ R
Arcsecant	sec-1(-x) = π -sec-1 (x), |x| ≥ 1
Arccosecant	cosec-1(-x) = -cosec-1 (x), |x| ≥ 1

Graphs of Inverse Trigonometric Functions 

Let us go through the six major types of inverse trigonometric functions, including their definitions, formulae, graphs, properties, and solved examples.

Arcsine

The arcsine function is the inverse of the sine function, which is represented as sin^-1 x. It is seen in the graph below:

Domain	-1 ≤ x ≤ 1
range	-π/2 ≤ y ≤ π/2

Arccosine

The arccosine function is the inverse of the cosine function, which is represented as cos^-1 x. It is seen in the graph below:

Domain	-1 ≤ x ≤ 1
range	0 ≤ y ≤ π

Arctangent

The arctangent function is the inverse of the tangent function, which is represented as tan^-1 x. It is seen in the graph below:

Domain	-∞ < x < ∞
range	-π/2 ≤ y ≤ π/2

Arccotangent

The arccotangent function is the inverse of the cotangent function, which is represented as cot^-1 x. It is seen in the graph below:

Domain	-∞ < x < ∞
range	0 ≤ y ≤ π

Arcsecant

The arcsecant function is the inverse of the secant function, which is represented as sec^-1 x. It is seen in the graph below:

Domain	-∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞
range	0 ≤ y ≤ π, y ≠ π/2

Arccosecant

The arccosecant function is the inverse of the cosecant function, which is represented as cosec^-1 x. It is seen in the graph below:

Domain	-∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞
range	-π/2 ≤ y ≤ π/2, y ≠ 0

Inverse Trigonometric Table

The notation, definition, domain and range of the basic inverse trigonometric function are given as:

Function Name	Notation	Definition	Domain of x	Range
Arcsine or inverse sine	y = sin-1 x	x = sin y	−1 ≤ x ≤ 1	− π/2 ≤ y ≤ π/2 -90°≤ y ≤ 90°
Arccosine or inverse cos	y = cos-1 x	x = cos y	−1 ≤ x ≤ 1	0 ≤ y ≤ π 0° ≤ y ≤ 180°
Arctangent or inverse tan	y = tan-1 x	x = tan y	For all real numbers	− π/2 < y < π/2 -90°< y < 90°
Arccotangent or inverse cot	y = cot-1 x	x = cot y	For all real numbers	0 < y < π 0° < y < 180°
Arcsecant or inverse secant	y = sec-1 x	x = sec y	x ≤ −1 or 1 ≤ x	0 ≤ y < π/2 or π/2 < y ≤ π 0° ≤ y < 90° or 90° < y ≤ 180°
Arccosecant or inverse cosecant	y = cosec-1 x	x = cosec y	x ≤ −1 or 1 ≤ x	−π/2 ≤ y < 0 or 0 < y ≤ π/2 −90° ≤ y < 0°or 0° < y ≤ 90°

Derivations of the Inverse Trigonometric Functions

It is important to keep the derivatives in mind to find the extremes of any inverse trigonometric function. Hence, the first-order derivatives of inverse trigonometric functions are as follows:

Inverse Trigonometric Functions	dy/dx
y = sin-1(x)	1 / √(1 – x2)
y = cos-1(x)	-1 / √(1 – x2)
y = tan-1(x)	1 / (1 + x2)
y = cot-1(x)	-1 / (1 + x2)
y = sec-1(x)	1 / [|x|√(x2 – 1)]
y = csc-1(x)	-1 / [|x|√(x2 – 1)]

Properties of Inverse Trigonometric Functions

Given below are a few properties of inverse trigonometric functions that you must practice thoroughly to score more, plus understand the essence of trigonometry better to apply the concept in real life. 

• Property set 01

sin⁻¹(x) = cosec⁻¹(1/x), x ∈ [−1,1] − {0}

cos⁻¹(x) = sec⁻¹(1/x), x ∈ [−1,1] − {0}

tan⁻¹(x) = cot⁻¹(1/x), if x > 0 or cot⁻¹(1/x) − π, if x < 0

cot⁻¹(x) = tan⁻¹(1/x), if x > 0 or tan⁻¹(1/x) + π, if x < 0

• Property set 02

sin⁻¹(−x) = −sin⁻¹(x)

tan⁻¹(−x) = −tan⁻¹(x)

cos⁻¹(−x) = π − cos⁻¹(x)

cosec⁻¹(−x) = − cosec⁻¹(x)

sec⁻¹(−x) = π − sec⁻¹(x)

cot⁻¹(−x) = π − cot⁻¹(x)

• Property set 03

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

cos⁻¹(1/x) = sec⁻¹x, x ≥ 1 or x ≤ −1

tan⁻¹(1/x) = −π + cot⁻¹(x)

• Property set 04

sin⁻¹(cos x) = π/2 − x, if x ∈ [0, π]

cos⁻¹(sin x) = π/2 − x, if x ∈ [−π/2, π/2]

tan⁻¹(cot x) = π/2 − x, x ∈ [0, π]

cot⁻¹(tan x) = π/2 − x, x ∈ [−π/2, π/2]

sec⁻¹(cosec x) = π/2 − x, x ∈ [−π/2, 0] ∪ [0, π/2]

cosec⁻¹(sec x) = π/2 − x, x ∈ [0, π] − {π/2}

sin⁻¹(x) = cos⁻¹[√(1−x²)], 0 ≤ x ≤ 1

• Property set 05

sin⁻¹ x + cos⁻¹ x = π/2

tan⁻¹ x + cot⁻¹ x = π/2

sec⁻¹ x + cosec⁻¹ x = π/2

• Property set 06

If x, y > 0

If x, y < 0

tan⁻¹(x) – tan⁻¹(y) = tan⁻¹[(x−y) / (1 + xy)], xy > −1

2 tan⁻¹(x) = tan⁻¹[(2x) / (1–x²)], |x|<1

• Property set 07

sin⁻¹(x) + sin⁻¹(y) = sin⁻¹[x  √(1 − y²) + y √(1 − x²)]

cos⁻¹x + cos⁻¹y = cos⁻¹[xy − √(1−x²)√(1 − y²)]

Let us understand Inverse Trigonometric Functions by some examples! 

Example 1: Find sin (cos⁻¹ 3/5)

Solution: Let, cos⁻¹ 3/5 = x 

Therefore, cos x = 3/5

sin x = √(1 – cos²  x) = √(1 – 9/25) = √(16/25) = 4/5 

Therefore, sin (cos⁻¹ 3/5) = sin x = 4/5.

Example 2: Find tan⁻¹ sin (- π/2)

Solution: tan⁻¹ sin (- π/2)

= tan⁻¹ (- sin π/2)

= tan⁻¹ (- 1)

= tan⁻¹(- tan π/4)

= tan⁻¹ tan (-π/4)

= – π/4. 

Therefore, tan⁻¹ sin (- π/2) = – π/4

Example 3: What will be the answer for sin⁻¹ (sin 10)?

Solution: sin⁻¹ (sin x) = x, if – π² ≤ x ≤ π².

x = 10 radians does not lie between – π² and π². But 3π – x i.e., 3π – 10 lies between – π² and π² and sin (3π – 10) = sin 10.

sin⁻¹ (sin 10)

= sin^-1 (sin (3π – 10)

= 3π – 10

Example 4: Find cos (tan⁻¹ ¾)

Solution: Let, tan⁻¹ ¾ = x 

tan x = ¾

sec² x – tan² x = 1

⇒ sec x = √(1 + tan² x)

⇒ sec x = √(1 + (3/4)²)

⇒ sec x = √(1 + 9/16)

⇒ sec x = √(25/16)

⇒ sec x = 5/4

Therefore, cos x = 4/5

⇒ x = cos⁻¹ 4/5

cos (tan⁻¹ ¾) = cos (cos⁻¹ 4/5) = 4/5

Example 5: Find sec csc⁻¹ (2/√3)

Solution: sec csc⁻¹ (2/√3)

= sec csc⁻¹ (csc π/3)

= sec (csc⁻¹ (csc π/3))

= sec π/3

= 2

Example 6: Determine the domain of cos^-1 ([2 + sin x] /3).

Solution: The domain of y = cos^-1 x is -1 ≤ x ≤ 1 or |x| ≤ 1. 

-1 ≤ [2 + sin x]/3 ≤ 1, same as – 3 ≤ 2 + sin x ≤ 3.

– 5 ≤ sin x ≤ 1 reduces to -1 ≤ sin x ≤ 1, 

which is equal to

– sin-1 (1) ≤ x ≤ sin-1 (1) OR π/2 ≤ x ≤ π/2.

Frequently Asked Questions

1. How do you calculate inverse function?

Ans. To calculate the inverse of a function, you must first know what the inverse of a function is. The inverse of a function is the function that takes the same input and returns the same output but reverses all operations in between.

So, if you have a function that takes an input and multiplies it by 2, then subtracts 5, then divides it by 7, and finally adds 6 to get the output:

f(x) = x * 2 – 5 / 7 + 6

The inverse of this function would be:

g(x) = (x * 2 – 5 / 7 + 6) / (x * 2 – 5 / 7 + 6) = x

This equation is essentially telling you that if you put any number into g(x), it will spit out another number that is equal to f(x). If you put in a negative number for x though, g(-1) will give you a negative number as well!

2. What is the inverse function of (F)?

Ans. The inverse function of (F) is:

G(x) = 1/(1 + x)

3. How to find the formula for an inverse function?

Ans. The formula for an inverse function is the same as the original, with a negative sign in front of it.

Let’s say you want to find the inverse of the function f(x) = 2x + 4. Well, we know that f(-1) = -1 * 2(-1 + 4) = -2+4 = -2. So we can write down our equation like this:

-2 = f(-1)

Now let’s solve for x by setting each side equal to -2 and simplifying:

x = (-2)/(-2) * 2(1+4) = 1

So our final formula is:f(-1) = 1

4. How to determine inverse function?

Ans. To determine an inverse function, you have to look at the exact opposite of the given function.

For example, if you are given a function of f(x)=x^2+1, then the inverse function will be g(x)=x^2-1.

The easiest way to determine the inverse function is to use the fact that you can always find a value for x and y so that when you plug x into one equation, the other equation will equal 0.

In this case, we would plug in x=0 into both equations: f(0) = 1 and g(0) = -1. This means that g(-1) = 1 and f(-1) = -1.

5. What are the methods to find the inverse of a function?

Ans. There are a number of methods to find the inverse of a function.

The first method is to use the reciprocal property. This means that if you select the inverse of the function, it will be equal to 1/f(x) or f(-x).

Another way to find an inverse is to use logarithm tables or online calculators. When you use these tools, you’ll be able to find both an exponential and logarithmic function’s inverse.