 # Trigonometry Formulas For Class 12

Trigonometry Formulas for Class 12 are available here for students. Practice trigonometric problems based on the formulas given here. The formulas involve here all the trigonometric functions and trigonometric identities. Trigonometry is a branch of Mathematics, that involves the study of the relationship between angles and lengths of triangles. In Class 12 Maths, we come across a different aspect of trigonometry, which is inverse trigonometric functions. Here, we will learn the domain and range of trigonometric functions. The basic trigonometric functions are sin, cos, tan, cot, sec and cosec.

## List of Class 12 Trigonometry Formulas

CBSE Class 12 mathematics contains Inverse Trigonometric Functions. This chapter includes definitions, graphs and elementary properties of inverse trigonometric functions. Trigonometry formulas for class 12 play a critical role in these chapters. Hence, all trigonometry formulas are provided here.

### Basic Trigonometric Functions

• Sin θ = Opposite side of angle θ/Hypotenuse
• Cos θ = Adjacent side of angle θ/Hypotenuse
• Tan θ = Opposite side of angle θ/Adjacent side of angle θ
• Sec θ = Hypotenuse/Adjacent side of angle θ
• Cosec θ = Hypotenuse/Opposite side of angle θ
• Cot θ = Adjacent side of angle θ/Opposite side of angle θ

### Domain and Range of Trigonometric Functions

Here are the domain and range of basic trigonometric functions:

• 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)

### Properties of Inverse Trigonometric Functions

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

### Addition Properties of Inverse Trigonometry functions

• sin-1a + cos-1a = π/2, a ∈ [– 1, 1]
• tan-1a + cot-1a = π/2, a ∈ R
• cosec-1a + sec-1a = π/2, | a | ≥ 1
• tan-1a + tan-1 b = tan-1 [(a+b)/1-ab], ab<1
• tan-1a – tan-1 b = tan-1 [(a-b)/1+ab], ab>-1
• tan-1a – tan-1 b = π + tan-1[(a+b)/1-ab], ab > 1; a,b > 0

### Twice of Inverse of Tan Function

• 2tan-1a = sin-1 [2a/(1+a2)], |a| ≤ 1
• 2tan-1a = cos-1[(1-a2)/(1+a2)], a ≥ 0
• 2tan-1a = tan-1[2a/(1+a2)], – 1 < a < 1

These are important formulas introduced in the Inverse trigonometric functions chapter of Class 12. Students can solve the problems based on these properties taking reference from this article.

### Video Lesson on Trigonometry ## Solved Examples

Q.1: What is the value of cot-1 (-1/√3)?

Solution: Suppose,

cot-1 (-1/√3) = x

cot x = -1/√3

Since,

cot (π/3) = 1/√3

Thus,

– cot (π/3) = cot (π – π/3) = cot 2π/3

– 1/√3  = cot 2π/3

or

cot-1 (- 1/√3) = 2π/3

Q.2: What is the value of cos (sec–1 x + cosec–1 x) when | x | ≥ 1?

Solution: Given,

cos (sec–1 x + cosec–1 x)

By the formula, we know that;

cosec–1 x + sec–1 x = π/2

Thus,

cos (sec–1 x + cosec–1 x) = cos π/2 = 0

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