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Chapter Analysis
Intermediate15 pages • EnglishQuick Summary
Chapter 2 of Class 12 Mathematics Part 1 focuses on Inverse Trigonometric Functions. It begins by reviewing the concept of inverse functions and the conditions under which they exist, particularly for trigonometric functions. The chapter discusses how to restrict the domains and ranges of trigonometric functions to allow their inverses to be defined. Examples are provided to demonstrate the calculation of inverse trigonometric values and to illustrate the use of inverse trigonometric functions in calculus, specifying their importance in integral evaluation. Additionally, the chapter includes exercises to reinforce the concepts.
Key Topics
- •Inverse Functions
- •Trigonometric Functions
- •Principal Value Branch
- •Domain Restriction
- •Graphical Representation
- •Properties of Inverse Trigonometric Functions
- •Applications in Calculus
Learning Objectives
- ✓Understand and define inverse trigonometric functions.
- ✓Apply domain and range restrictions to define inverse functions.
- ✓Calculate and interpret the principal values of inverse trigonometric functions.
- ✓Graph inverse trigonometric functions and understand their properties.
- ✓Apply inverse trigonometric functions in calculus, particularly for integration.
Questions in Chapter
Find the principal value of sin–1(1/2).
Page 26
Prove that sin–1(22x - 1) = 2 sin–1(x), for -1/2 <= x <= 1/2.
Page 29
Prove that 3cos–1(x) = cos–1(4x^3 - 3x), for x ∈ [-1, 1].
Page 29
Find the value of tan–1(2cos(2sin–1(x))) + cos–1(2sin(x)).
Page 31
Additional Practice Questions
What is the principal value of sec–1(2)?
mediumAnswer: The principal value of sec–1(2) is pi/3, because secant is the reciprocal of cosine, and cos(pi/3) = 1/2.
Express cot–1(1) in terms of pi.
easyAnswer: cot–1(1) is equal to pi/4, because cotangent is the reciprocal of tangent, and tan(pi/4) = 1.
Evaluate tan–1(√3) + tan–1(1/√3).
mediumAnswer: tan–1(√3) = pi/3 and tan–1(1/√3) = pi/6. Their sum equals pi/2.
If y = sin–1(x), find the derivative dy/dx.
hardAnswer: The derivative dy/dx = 1/√(1-x^2), where |x| < 1.
Show that sin–1(x) + cos–1(x) = pi/2 for all x in [-1, 1].
easyAnswer: For any x in [-1, 1], sin–1(x) and cos–1(x) are complementary angles, hence their sum is pi/2.