CBSE Class 12 Maths Chapter-2 Inverse Trigonometric Functions Formula

Q: 1. What is meant by Inverse Trigonometric Function?

The inverse trigonometric functions are called anti trigonometric functions or sometimes known as the cyclometric functions or arcus functions. The inverse trigonometric functions of sine, cosine, tangent, secant, cosecant, and cotangent are used to determine the angle of a triangle from any of the trigonometric functions.

Q: 2. What are the applications of Inverse Trigonometric Function?

It is widely used in different fields including geometry, physics, engineering etc. But in general, the convention symbol to indicate the inverse trigonometric function with the help of arc-prefix such as arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). In order to identify the sides of a triangle when the remaining side lengths are known.Take into the function y = f(x), and x = g(y) then the inverse function is expressed as g = f-1,This implies that if y=f(x), then x = f-1(y).In a way that f(g(y))=y and g(f(y))=x.Example of Inverse trigonometric functions: x= sin-1y

Q: 3. What are the properties of inverse trigonometric functions?

Below is the inverse trigonometric function formula:sin-1 (1/a) = cosec-1(a), a ≥ 1 or a ≤ – 1sin-1(–a) = – sin-1(a), a ∈ [– 1, 1]cos-1(1/a) = sec-1(a), a ≥ 1 or a ≤ – 1cos-1(–a) = π – cos-1(a), a ∈ [– 1, 1]tan-1(1/a) = cot-1(a), a>0tan-1(–a) = – tan-1(a), a ∈ Rsec-1(–a) = π – sec-1(a), | a | ≥ 1cosec-1(–a) = –cosec-1(a), | a | ≥ 1cot-1(–a) = π – cot-1(a), a ∈ R

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CBSE Class 12 Maths Chapter-2 Inverse Trigonometric Functions Formula

Last updated date: 25th Apr 2024

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Inverse Trigonometric Functions Formula for CBSE Class 12 Maths - Free PDF Download

Inverse Trigonometric Formulas are part of Trigonometry which is further an element of geometry. Inverse trigonometric functions formulas enable us to learn about the association between sides and angles of a right-angled triangle. In Class 11 and 12, you will come across inverse trigonometric functions formulas list Class 12 PDF, based on the ratios and functions such as sin, cos, and tan. In the same manner, the inverse trigonometric functions are expressed as sin-1 x, cos-1 x, tan-1 x, cot-1 x, sec-1 x, cosec-1 x.

Inverse Trigonometric Functions Formulas List

Now, let us get the formulas linked with these functions. In order to solve different types of inverse trigonometric functions, inverse trigonometry formulas are extracted from some basic properties of trigonometry. Inverse trigonometric function formula list is provided below for reference to solve the problems.

Inverse Trigonometric Functions Formulas

S.No	Inverse Trigonometry Class 12 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	cot-1(-x) = π – cot-1(x), x ∈ R
5	sec-1(-x) = π -sec-1(x), \|x\| ≥ 1
6	cosec-1(-x) = -cosec-1(x), \|x\| ≥ 1
7	sin-1x + cos-1x = π/2 , x ∈ [-1, 1]
8	sin-1(1/x) = cosec-1(x), if x ≥ 1 or x ≤ -1
9	cos-1(1/x) = sec-1(x), if x ≥ 1 or x ≤ -1
10	tan-1x + cot-1x = π/2 , x ∈ R
11	tan-1(1/x) = cot-1(x), x > 0
12	tan-1 x + tan-1 y = tan-1((x+y)/(1-xy)), if the value xy < 1
13	tan-1 x – tan-1 y = tan-1((x-y)/(1+xy)), if the value xy > -1
14	2 tan-1 x = sin-1(2x/(1+x2)), \|x\| ≤ 1
15	sec-1x + cosec-1x = π/2 ,\|x\| ≥ 1
16	3sin-1x = sin-1(3x-4x3)
17	sin(sin-1(x)) = x, -1≤ x ≤1
18	3cos-1x = cos-1(4x3-3x)
19	cos(cos-1(x)) = x, -1≤ x ≤1
20	3tan-1x = tan-1((3x-x3)/(1-3x2)
21	tan(tan-1(x)) = x, – ∞ < x < ∞
22	sec(sec-1(x)) = x,- ∞ < x ≤ 1 or 1 ≤ x < ∞
23	cosec(cosec-1(x)) = x, – ∞ < x ≤ 1 or -1 ≤ x < ∞
24	cot(cot-1(x)) = x, – ∞ < x < ∞
25	sin-1(sin θ) = θ, -π/2 ≤ θ ≤π/2
26	cos-1(cos θ) = θ, 0 ≤ θ ≤ π
27	tan-1(tan θ) = θ, -π/2 < θ < π/2
28	sec-1(sec θ) = θ, 0 ≤ θ ≤ π/2 or π/2< θ ≤ π
29	cosec-1(cosec θ) = θ, – π/2 ≤ θ < 0 or 0 < θ ≤ π/2
30	cot-1(cot θ) = θ, 0 < θ < π

Inverse Trigonometric Formulas Class 12

List class CBSE Class 12 Trigonometry Formulas mathematics contains Inverse Trigonometric formulas for Class 12. This chapter includes the definition, inverse trigonometry formula, graphs and elementary properties of inverse trigonometric functions. Trigonometry Formulas for Class 12 play a key role in these chapters. Thus, all trigonometry formulas are given here.

Basic Concepts Included in Trigonometry Formulas For Class 12

Below are the basic trigonometric functions based on the domain and range

Basic Trigonometric Functions Based on the Domain and Range

Trigonometric Functions	Inverse Trigonometric Formulas
Sine Function	sine: R → [– 1, 1]
Cosine Function	cos : R → [– 1, 1]
Tangent Function	tan : R – { x : x = (2n + 1) π/2, n ∈ Z} →R
Cotangent Function	cot : R – { x : x = nπ, n ∈ Z} →R
Secant Function	Sec: R – { x : x = (2n + 1) π/2, n ∈ Z} →R – (– 1, 1)
Cosecant Function	cosec: R – { x : x = nπ, n ∈ Z} →R – (– 1, 1)

FAQs on CBSE Class 12 Maths Chapter-2 Inverse Trigonometric Functions Formula

1. What is meant by Inverse Trigonometric Function?

The inverse trigonometric functions are called anti trigonometric functions or sometimes known as the cyclometric functions or arcus functions. The inverse trigonometric functions of sine, cosine, tangent, secant, cosecant, and cotangent are used to determine the angle of a triangle from any of the trigonometric functions.

2. What are the applications of Inverse Trigonometric Function?

It is widely used in different fields including geometry, physics, engineering etc. But in general, the convention symbol to indicate the inverse trigonometric function with the help of arc-prefix such as arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). In order to identify the sides of a triangle when the remaining side lengths are known.

Take into the function y = f(x), and x = g(y) then the inverse function is expressed as g = f^-1,

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

In a way that f(g(y))=y and g(f(y))=x.

Example of Inverse trigonometric functions: x= sin^-1y

3. What are the properties of inverse trigonometric functions?

Below is the inverse trigonometric function formula:

sin^-1 (1/a) = cosec^-1(a), a ≥ 1 or a ≤ – 1
sin^-1(–a) = – sin^-1(a), a ∈ [– 1, 1]
cos^-1(1/a) = sec^-1(a), a ≥ 1 or a ≤ – 1
cos^-1(–a) = π – cos^-1(a), a ∈ [– 1, 1]
tan^-1(1/a) = cot^-1(a), a>0
tan^-1(–a) = – tan^-1(a), a ∈ R
sec^-1(–a) = π – sec^-1(a), | a | ≥ 1
cosec^-1(–a) = –cosec^-1(a), | a | ≥ 1
cot^-1(–a) = π – cot^-1(a), a ∈ R