Questions
3 questions
Difficulty
Medium
Importance
Medium yield, essential for calculus
Overview
Inverse Trigonometric Functions represent the inverse operation of trigonometric functions, essential for solving equations involving angles. Mastering the domain and range restrictions is critical for determining principal values, which is a frequent testing point in Class 12 board examinations. A clear understanding of these properties allows for the simplification of complex inverse expressions.
Principal Value Branches
The inverse trigonometric functions are multi-valued; however, we restrict their domains to obtain unique values called principal values. Always verify that your final answer falls within the specified principal value branch before concluding your calculation.
- sin^-1(x) domain: [-1, 1], range: [-π/2, π/2]
- cos^-1(x) domain: [-1, 1], range: [0, π]
- tan^-1(x) domain: R, range: (-π/2, π/2)
- cosec^-1(x) domain: R - (-1, 1), range: [-π/2, π/2] - {0}
- sec^-1(x) domain: R - (-1, 1), range: [0, π] - {π/2}
- cot^-1(x) domain: R, range: (0, π)
Properties of Inverse Functions
These properties allow for the direct conversion between trigonometric and inverse functions. They are fundamental for solving equations where variables are nested inside inverse operations.
- sin^-1(-x) = -sin^-1(x)
- cos^-1(-x) = π - cos^-1(x)
- tan^-1(-x) = -tan^-1(x)
- sin^-1(x) + cos^-1(x) = π/2
- tan^-1(x) + cot^-1(x) = π/2
- cosec^-1(x) + sec^-1(x) = π/2
Conversion and Addition Identities
Most complex problems require converting all inverse functions into a single type (usually tan^-1) or using addition/subtraction identities to condense expressions. These identities are the most common source of high-mark questions in the board exams.
- tan^-1(x) + tan^-1(y) = tan^-1((x+y)/(1-xy))
- tan^-1(x) - tan^-1(y) = tan^-1((x-y)/(1+xy))
- 2tan^-1(x) = sin^-1(2x/(1+x^2))
- 2tan^-1(x) = cos^-1((1-x^2)/(1+x^2))
- 2tan^-1(x) = tan^-1(2x/(1-x^2))
Formula Sheet
sin^-1(x) = cosec^-1(1/x)
cos^-1(x) = sec^-1(1/x)
tan^-1(x) = cot^-1(1/x)
sin^-1(-x) = -sin^-1(x)
cos^-1(-x) = π - cos^-1(x)
tan^-1(x) + tan^-1(y) = tan^-1((x+y)/(1-xy))
2tan^-1(x) = sin^-1(2x/(1+x^2)) = cos^-1((1-x^2)/(1+x^2)) = tan^-1(2x/(1-x^2))
Exam Tip
When simplifying complex inverse expressions, always substitute x = tanθ to transform the expression into a standard trigonometric identity.
Common Mistakes
- Neglecting the domain restrictions, especially for negative arguments in sec^-1, cosec^-1, and cot^-1.
- Assuming sin^-1(x) is the same as (sin x)^-1; they are fundamentally different functions.
- Forgetting to check if the product xy < 1 before applying the standard tan^-1(x) + tan^-1(y) formula.
More Revision Notes
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