Inverse Trigonometric Formula PDF, Check Inverse Trigonometric Functions Table
Anchal Singh26 Aug, 2025
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Inverse trigonometric formulas are special mathematical expressions used to find the angle when the value of a trigonometric function like sine, cosine, or tangent is known. These formulas help reverse the usual trigonometric functions and are important in solving problems related to triangles, calculus, and physics.
Knowing these formulas makes it easier for students, especially those in Class 12, to solve questions involving angles and side ratios efficiently.
Inverse Trigonometric Formulas
Inverse trigonometric functions formulas are the inverse operations of the basic trigonometric functions, such as sine, cosine, and tangent.
While the regular trigonometric functions find the ratio of sides in a right triangle given an angle, inverse trigonometric functions help find the angle when the ratio is known.
Consider, the function y = f(x), and x = g(y) then the inverse function is written as g = f-1,
This means that if y=f(x), then x = f-1(y).
Such that f(g(y))=y and g(f(y))=x.
Example of Inverse trigonometric functions: x= sin-1y
Below is the list of inverse trigonometric functions along with their domain and range values, which is essential knowledge in the inverse trigonometric formulas class 12 syllabus:
Inverse Trigonometric Formulas List
Below is a list of important inverse trigonometric formulas that will help you solve various problems easily. These formulas cover basic properties, addition and subtraction rules, and principal value relationships of inverse trig functions.
|
S.No |
Inverse Trigonometric Formulas |
|
1 |
sin-1(-x) = -sin-1(x), x ∈ [-1, 1] |
|
2 |
cos-1(-x) = π -cos-1(x), x ∈ [-1, 1] |
|
3 |
tan-1(-x) = -tan-1(x), x ∈ R |
|
4 |
cosec-1(-x) = -cosec-1(x), |x| ≥ 1 |
|
5 |
sec-1(-x) = π -sec-1(x), |x| ≥ 1 |
|
6 |
cot-1(-x) = π – cot-1(x), x ∈ R |
|
7 |
sin-1x + cos-1x = π/2 , x ∈ [-1, 1] |
|
8 |
tan-1x + cot-1x = π/2 , x ∈ R |
|
9 |
sec-1x + cosec-1x = π/2 ,|x| ≥ 1 |
|
10 |
sin-1(1/x) = cosec-1(x), if x ≥ 1 or x ≤ -1 |
|
11 |
cos-1(1/x) = sec-1(x), if x ≥ 1 or x ≤ -1 |
|
12 |
tan-1(1/x) = cot-1(x), x > 0 |
|
13 |
tan-1 x + tan-1 y = tan-1((x+y)/(1-xy)), if the value xy < 1 |
|
14 |
tan-1 x – tan-1 y = tan-1((x-y)/(1+xy)), if the value xy > -1 |
|
15 |
2 tan-1 x = sin-1(2x/(1+x2)), |x| ≤ 1 |
|
16 |
2tan-1 x = cos-1((1-x2)/(1+x2)), x ≥ 0 |
|
17 |
2tan-1 x = tan-1(2x/(1-x2)), -1<x<1 |
|
18 |
3sin-1x = sin-1(3x-4x3) |
|
19 |
3cos-1x = cos-1(4x3-3x) |
|
20 |
3tan-1x = tan-1((3x-x3)/(1-3x2)) |
|
21 |
sin(sin-1(x)) = x, -1≤ x ≤1 |
|
22 |
cos(cos-1(x)) = x, -1≤ x ≤1 |
|
23 |
tan(tan-1(x)) = x, – ∞ < x < ∞. |
|
24 |
cosec(cosec-1(x)) = x, – ∞ < x ≤ 1 or -1 ≤ x < ∞ |
|
25 |
sec(sec-1(x)) = x,- ∞ < x ≤ 1 or 1 ≤ x < ∞ |
|
26 |
cot(cot-1(x)) = x, – ∞ < x < ∞. |
|
27 |
sin-1(sin θ) = θ, -π/2 ≤ θ ≤π/2 |
|
28 |
cos-1(cos θ) = θ, 0 ≤ θ ≤ π |
|
29 |
tan-1(tan θ) = θ, -π/2 < θ < π/2 |
|
30 |
cosec-1(cosec θ) = θ, – π/2 ≤ θ < 0 or 0 < θ ≤ π/2 |
|
31 |
sec-1(sec θ) = θ, 0 ≤ θ ≤ π/2 or π/2< θ ≤ π |
|
32 |
cot-1(cot θ) = θ, 0 < θ < π |
Inverse Trigonometric Functions Table
The Inverse Trigonometric Functions Table helps in finding angles corresponding to given trigonometric values.
It is widely used in solving equations, integrals, and higher-level mathematics problems.
| Function Name | Notation | Definition | Domain of x | Range |
| Arcsine or inverse sine | y = sin -1 (x) | x=sin y | −1 ≤ x ≤ 1 |
|
| Arccosine or inverse cosine | y=cos -1 (x) | x=cos y | −1 ≤ x ≤ 1 |
|
| Arctangent or inverse tangent | y=tan -1 (x) | x=tan y | For all real numbers |
|
| Arccotangent or inverse cot | y=cot -1 (x) | x=cot y | For all real numbers |
|
| Arcsecant or inverse secant | y = sec -1 (x) | x=sec y | x ≤ −1 or 1 ≤ x |
|
| Arccosecant | y=csc -1 (x) | x=csc y | x ≤ −1 or 1 ≤ x |
|
Derivative Formula of Inverse Trigonometric Functions
For calculus students, knowing the derivative formula of inverse trigonometric functions is essential. Some important derivatives are:
Integration Formulas of Inverse Trigonometric Functions
In calculus, integration involving inverse trig functions can be solved using the following integration formulas of inverse trigonometric functions:
Inverse Trigonometric Functions Properties
The inverse trigonometric functions, often referred to as arc functions, are defined within specific intervals or under restricted domains. You can find more information about the properties of inverse trigonometric functions here.
Also Check - Probability Formula
Fundamental trigonometry concepts encompass essential trigonometric ratios and functions, including sine (sin x), cosine (cos x), tangent (tan x), cosecant (csc x), secant (sec x), and cotangent (cot x).
Also Check - Inverse Matrix Formula
Inverse Trigonometric Functions Problems
Example 1: Determine x for which sin(x) equals 2.
Solution: Given: sin x = 2
x = sin^(-1)(2), which is not feasible.
Thus, there exists no x for which sin x equals 2. Therefore, the domain of sin^(-1)x is limited to values of x between -1 and 1.
Example 2: Calculate sin^(-1)(sin(π/6)).
Solution:
sin^(-1)(sin(π/6)) = π/6 (Utilizing the identity sin^(-1)(sin(x)) = x)
Also Check - Linear Inequalities Formula
Download Inverse Trigonometric Formulas PDF
To make learning easier, students can download an inverse trigonometric formulas PDF. This PDF compiles all inverse trigonometric formulas in one place for quick revision, especially helpful for Class 12 board exams and competitive tests.
Inverse Trigonometric Formula FAQs
What is an inverse trigonometric function?
Why are inverse trigonometric formulas important?
Where are inverse trigonometric functions used?
What are the domain and range of inverse trigonometric functions?
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