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:

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:

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:

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:

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:

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:

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:

Inverse Trigonometric Table

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

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:

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.