Inverse Trigonometric Functions

Inverse trigonometric functions answer the reverse question: if sin(θ) = 0.5, what is θ? While regular trigonometric functions take angles and give us ratios (sin, cos, tan), inverse trig functions take ratios and give us angles back. But there's a twist: trigonometric functions aren't one-one over their entire domains, so we must carefully restrict their domains to make inverses possible. This chapter explores these restrictions and the remarkable properties of arcsin, arccos, and arctan.

The Problem: Why Inverses Don't Exist Naturally

Recall from chapter-01-relations-and-functions that a function can only have an inverse if it's one-one and onto. Consider sin(x): the value sin(π/6) = 0.5, but also sin(5π/6) = 0.5. The same output comes from different inputs, so sin is not one-one over all real numbers.

This is where domain restriction becomes crucial. By limiting sin(x) to [-π/2, π/2], we capture all possible output values [-1, 1] exactly once. Now the inverse function arcsin (or sin⁻¹) exists with domain [-1, 1] and range [-π/2, π/2].

Understanding Each Inverse Trig Function

Arcsine: arcsin(x) or sin⁻¹(x)

• Domain: [-1, 1]
• Range: [-π/2, π/2]
• Relationship: If y = arcsin(x), then sin(y) = x and -π/2 ≤ y ≤ π/2
• Intuition: "Find the angle between -90° and 90° whose sine is x"

Arccosine: arccos(x) or cos⁻¹(x)

• Domain: [-1, 1]
• Range: [0, π]
• Relationship: If y = arccos(x), then cos(y) = x and 0 ≤ y ≤ π
• Intuition: "Find the angle between 0° and 180° whose cosine is x"

Arctangent: arctan(x) or tan⁻¹(x)

• Domain: All real numbers ℝ
• Range: (-π/2, π/2)
• Relationship: If y = arctan(x), then tan(y) = x and -π/2 < y < π/2
• Intuition: "Find the angle whose tangent is x" (tan can produce any real value)

Building from Class 11

In Class 11, you learned that sin, cos, and tan functions are periodic and not one-one. You also learned that inverse relations exist, but they don't form functions unless we restrict domains. This chapter makes those inverses into proper functions through careful domain restrictions.

Key Properties and Relationships

Fundamental Identities

• sin(arcsin(x)) = x for all x ∈ [-1, 1]
• arcsin(sin(θ)) = θ only if θ ∈ [-π/2, π/2]
• cos(arccos(x)) = x for all x ∈ [-1, 1]
• tan(arctan(x)) = x for all x ∈ ℝ

Complementary Relationships

• arcsin(x) + arccos(x) = π/2 for all x ∈ [-1, 1]
• This reveals deep symmetry: if arcsin(x) is small, arccos(x) is large, and they always sum to π/2

Important Derivative Relationships

These lead directly to chapter-05-continuity-and-differentiability:

• d/dx[arcsin(x)] = 1/√(1 - x²)
• d/dx[arccos(x)] = -1/√(1 - x²)
• d/dx[arctan(x)] = 1/(1 + x²)

Graphical Interpretations

The graphs of inverse trig functions are reflections of their parent functions across the line y = x. For arcsin(x): plot sin(x) on [-π/2, π/2], then flip it about y = x to get a curve rising from (-1, -π/2) to (1, π/2).

Connections to Other Topics

• chapter-01-relations-and-functions: Domain restrictions make these relations into proper functions
• chapter-05-continuity-and-differentiability: Derivatives of inverse trig functions and their differentiability
• chapter-06-application-of-derivatives: Using inverse trig functions to solve optimization problems
• Class 11 Trigonometry: Parent functions and their properties

Socratic Questions

1. Why must we restrict the domain of sin(x) to [-π/2, π/2] rather than some other interval like [0, π] to create its inverse function? What property must a domain restriction maintain?

1. Consider the equation sin(θ) = 0.5. Without using inverse trig functions, how many solutions exist in [0, 2π)? How many solutions would exist if we used θ = arcsin(0.5)? Why the difference?

1. The domain of arctan(x) is all real numbers, while arcsin(x) and arccos(x) have limited domains. Why can arctangent accept any real input, while arcsine and arccosine cannot?

1. If you know arcsin(x) + arccos(x) = π/2, can you explain geometrically why complementary angles sum to π/2? What does this tell you about the relationship between sine and cosine?

1. When solving a problem that yields arcsin(2), why would this be impossible? What does this tell us about the relationship between the range of trigonometric functions and the domain of their inverses?