Class 12 Maths Inverse Trigonometric Functions Notes

2.1 Introduction

Trigonometric functions such as sin, cos, tan, cot, sec, and cosec are widely used in mathematics. However, these functions are not one-one over their complete domains, so their inverses do not exist directly.

To define their inverses, we restrict the domain of trigonometric functions so that they become one-one and onto. The inverse of these restricted functions are called inverse trigonometric functions.

Inverse trigonometric functions are useful in calculus, coordinate geometry, and real-life applications involving angles.

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2.2 Basic Concepts

If a function f has an inverse f⁻¹, then:$$f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x$$f(f−1(x))=xandf−1(f(x))=x

🔹 Inverse Trigonometric Functions

Trigonometric Function	Inverse Function
sin x	sin⁻¹ x (arcsin x)
cos x	cos⁻¹ x (arccos x)
tan x	tan⁻¹ x (arctan x)
cot x	cot⁻¹ x
sec x	sec⁻¹ x
cosec x	cosec⁻¹ x

Note: sin⁻¹x does NOT mean 1/sin x. It means the angle whose sine is x.

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🔹 Principal Value Branch

To make trigonometric functions invertible, their domains are restricted as follows:

Function	Restricted Domain	Range
sin x	$[-\frac{\pi}{2}, \frac{\pi}{2}]$[−2π​,2π​]	$[-1, 1]$[−1,1]
cos x	$[0, \pi]$[0,π]	$[-1, 1]$[−1,1]
tan x	$(-\frac{\pi}{2}, \frac{\pi}{2})$(−2π​,2π​)	ℝ
cot x	$(0, \pi)$(0,π)	ℝ
sec x	$[0, \pi] \setminus \{\frac{\pi}{2}\}$[0,π]∖{2π​}	$(-\infty, -1] \cup [1, \infty)$(−∞,−1]∪[1,∞)
cosec x	$[-\frac{\pi}{2}, \frac{\pi}{2}] \setminus \{0\}$[−2π​,2π​]∖{0}	$(-\infty, -1] \cup [1, \infty)$(−∞,−1]∪[1,∞)

The selected domain is called the principal value branch.

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2.3 Properties of Inverse Trigonometric Functions

🔹 1. Domain and Range

• Domain of sin⁻¹x and cos⁻¹x is [-1, 1]
• Domain of tan⁻¹x, cot⁻¹x is ℝ
• Range depends on the principal value branch

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🔹 2. Fundamental Properties

1. sin(sin⁻¹x) = x, for x ∈ [-1, 1]
2. cos(cos⁻¹x) = x, for x ∈ [-1, 1]
3. tan(tan⁻¹x) = x, for all real x

But,

• sin⁻¹(sin x) ≠ x for all x
• cos⁻¹(cos x) ≠ x for all x

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🔹 3. Important Identities

• sin⁻¹x + cos⁻¹x = π/2
• tan⁻¹x + cot⁻¹x = π/2
• sec⁻¹x + cosec⁻¹x = π/2

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🔹 4. Negative Argument Properties

• sin⁻¹(-x) = −sin⁻¹x
• tan⁻¹(-x) = −tan⁻¹x
• cos⁻¹(-x) = π − cos⁻¹x

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✅ Key Points to Remember

• Inverse trigonometric functions exist only after restricting domains.
• sin⁻¹x, cos⁻¹x, etc. represent angles, not ratios.
• Principal value branches are very important for solving problems.
• These concepts are heavily used in limits, differentiation, and integration.