Chapter 2Mathematics - Part I

Inverse Trigonometric Functions

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Inverse Trigonometric Functions

Inverse Trigonometric Functions

Chapter Overview

Inverse Trigonometric Functions deal with the reverse process of trigonometric functions. These functions are used to obtain an angle from any of the trigonometric ratios. Inverse trigonometric functions are used in various fields like physics, engineering, and mathematics. The chapter will cover the definition, range, domain, and properties of inverse trigonometric functions.

Learning Objectives

  • Define the inverse trigonometric functions.
  • Determine the range and domain of inverse trigonometric functions.
  • Understand the properties of inverse trigonometric functions.
  • Solve problems involving inverse trigonometric functions.

Important Concepts

Definition of Inverse Trigonometric Functions

Inverse trigonometric functions are used to obtain an angle from any of the trigonometric ratios. These functions are denoted by the following symbols:

  • arcsin(x) or sin^(-1)(x)
  • arccos(x) or cos^(-1)(x)
  • arctan(x) or tan^(-1)(x)
  • arccot(x) or cot^(-1)(x)
  • arcsec(x) or sec^(-1)(x)
  • arccsc(x) or csc^(-1)(x)

Inverse trigonometric functions are used to find the angle whose trigonometric ratio is given. For example, if we are given the value of the sine of an angle, we can use the inverse sine function to find the angle.

Range and Domain of Inverse Trigonometric Functions

  • The range of arcsin(x) is [-π/2, π/2] and its domain is [-1, 1].
  • The range of arccos(x) is [0, π] and its domain is [-1, 1].
  • The range of arctan(x) is (-π/2, π/2) and its domain is all real numbers.
  • The range of arccot(x) is (0, π) and its domain is all real numbers except 0.
  • The range of arcsec(x) is [0, π/2) ∪ (π/2, π] and its domain is all real numbers except -1 and 1.
  • The range of arccsc(x) is [-π/2, 0) ∪ (0, π/2] and its domain is all real numbers except -1 and 1.

The range and domain of inverse trigonometric functions are different from those of trigonometric functions. The range of an inverse trigonometric function is the set of all possible output values, while the domain is the set of all possible input values.

Properties of Inverse Trigonometric Functions

  • The inverse trigonometric functions are one-to-one functions.
  • The inverse trigonometric functions are continuous functions.
  • The inverse trigonometric functions are differentiable functions.

One-to-one functions have a unique output value for each input value. Continuous functions have no gaps or jumps in their graphs. Differentiable functions have derivatives that exist at all points.

Composition of Inverse Trigonometric Functions

  • The composition of inverse trigonometric functions is used to obtain an angle from any of the trigonometric ratios.
  • The composition of inverse trigonometric functions is used in various fields like physics, engineering, and mathematics.

Composition of functions means applying one function to the output of another function. Inverse trigonometric functions can be composed to find the angle whose trigonometric ratio is given.

Advanced Concepts

Deep-Dive Case Studies and Real-Life Applications

Inverse trigonometric functions are used in various fields like physics, engineering, and mathematics. They are used to obtain an angle from any of the trigonometric ratios.

Case Study 1: Inverse trigonometric functions are used in navigation systems to find the angle of elevation of a satellite.

Case Study 2: Inverse trigonometric functions are used in medical imaging to reconstruct images of the body.

Real-Life Application 1: Inverse trigonometric functions are used in GPS systems to find the angle of elevation of a satellite.

Real-Life Application 2: Inverse trigonometric functions are used in medical imaging to reconstruct images of the body.

Step-by-Step Problem Solving Strategies & Detailed Proofs

To solve problems involving inverse trigonometric functions, follow these steps:

  1. Identify the type of inverse trigonometric function required.
  2. Determine the range and domain of the inverse trigonometric function.
  3. Use the inverse trigonometric function to find the angle.
  4. Check the domain and range of the inverse trigonometric function.

Detailed Proof 1: To prove that arcsin(x) is a one-to-one function, we need to show that it has a unique output value for each input value.

Let x and y be two input values such that arcsin(x) = arcsin(y). Then, sin(arcsin(x)) = sin(arcsin(y)).

Since sin(x) is a one-to-one function, we have x = y.

Therefore, arcsin(x) is a one-to-one function.

Detailed Proof 2: To prove that arccos(x) is a continuous function, we need to show that it has no gaps or jumps in its graph.

Let x be an input value and let {x_n} be a sequence of input values that converges to x.

Then, arccos(x_n) converges to arccos(x).

Therefore, arccos(x) is a continuous function.

Higher-Order Thinking Skills (HOTS) Questions

  1. Prove that arcsin(x) is a one-to-one function.
  2. Prove that arccos(x) is a continuous function.
  3. Find the angle whose trigonometric ratio is given by sin(π/4) = 1/√2.
  4. Find the angle whose trigonometric ratio is given by cos(π/3) = 1/2.
  5. Find the angle whose trigonometric ratio is given by tan(π/4) = 1.

Previous Year Questions (PYQs) with Solutions

  1. Find the value of arcsin(1/√2). Solution: arcsin(1/√2) = π/4.
  2. Find the value of arccos(1/2). Solution: arccos(1/2) = π/3.
  3. Find the value of arctan(1). Solution: arctan(1) = π/4.
  4. Find the value of arccot(1). Solution: arccot(1) = π/4.
  5. Find the value of arcsec(1). Solution: arcsec(1) = π/4.

NCERT Textbook Questions & Detailed Answers

Exercise 1.1

  1. Find the value of arcsin(1/2). Solution: arcsin(1/2) = π/6.
  2. Find the value of arccos(√3/2). Solution: arccos(√3/2) = π/6.
  3. Find the value of arctan(1/√3). Solution: arctan(1/√3) = π/6.
  4. Find the value of arccot(√3). Solution: arccot(√3) = π/6.
  5. Find the value of arcsec(2). Solution: arcsec(2) = π/3.

Exercise 1.2

  1. Find the value of arcsin(-1/2). Solution: arcsin(-1/2) = -π/6.
  2. Find the value of arccos(-√3/2). Solution: arccos(-√3/2) = 5π/6.
  3. Find the value of arctan(-1/√3). Solution: arctan(-1/√3) = -π/6.
  4. Find the value of arccot(-√3). Solution: arccot(-√3) = -π/6.
  5. Find the value of arcsec(-2). Solution: arcsec(-2) = -π/3.

Exercise 1.3

  1. Find the value of arcsin(-1). Solution: arcsin(-1) = -π/2.
  2. Find the value of arccos(-1). Solution: arccos(-1) = π.
  3. Find the value of arctan(-1). Solution: arctan(-1) = -π/4.
  4. Find the value of arccot(-1). Solution: arccot(-1) = -π/4.
  5. Find the value of arcsec(-1). Solution: arcsec(-1) = -π/2.

Exercise 1.4

  1. Find the value of arcsin(1). Solution: arcsin(1) = π/2.
  2. Find the value of arccos(1). Solution: arccos(1) = 0.
  3. Find the value of arctan(1). Solution: arctan(1) = π/4.
  4. Find the value of arccot(1). Solution: arccot(1) = π/4.
  5. Find the value of arcsec(1). Solution: arcsec(1) = π/2.

Exercise 1.5

  1. Find the value of arcsin(0). Solution: arcsin(0) = 0.
  2. Find the value of arccos(0). Solution: arccos(0) = π/2.
  3. Find the value of arctan(0). Solution: arctan(0) = 0.
  4. Find the value of arccot(0). Solution: arccot(0) = π/2.
  5. Find the value of arcsec(0). Solution: arcsec(0) = π/2.

Exercise 1.6

  1. Find the value of arcsin(-√3/2). Solution: arcsin(-√3/2) = -π/3.
  2. Find the value of arccos(√3/2). Solution: arccos(√3/2) = π/3.
  3. Find the value of arctan(-√3). Solution: arctan(-√3) = -π/3.
  4. Find the value of arccot(-√3). Solution: arccot(-√3) = -π/3.
  5. Find the value of arcsec(-√3). Solution: arcsec(-√3) = -π/3.

Exercise 1.7

  1. Find the value of arcsin(1/√2). Solution: arcsin(1/√2) = π/4.
  2. Find the value of arccos(1/√2). Solution: arccos(1/√2) = π/4.
  3. Find the value of arctan(1/√3). Solution: arctan(1/√3) = π/6.
  4. Find the value of arccot(√3). Solution: arccot(√3) = π/6.
  5. Find the value of arcsec(2). Solution: arcsec(2) = π/3.

Exercise 1.8

  1. Find the value of arcsin(-1/√2). Solution: arcsin(-1/√2) = -π/4.
  2. Find the value of arccos(-1/√2). Solution: arccos(-1/√2) = 3π/4.
  3. Find the value of arctan(-1/√3). Solution: arctan(-1/√3) = -π

Pro Tip for this Chapter

Ensure you practice the in-text questions provided in the official NCERT PDF. If you find any topic difficult, review the formulas and concepts highlighted above. For advanced doubts, join our classroom coaching in Begusarai.