Mathematics Class 12 - Inverse Trigonometric-Functions Notes

Inverse Trigonometric Functions

Welcome to the chapter on Inverse Trigonometric Functions for Class 12. In this chapter, you will learn about the inverse of trigonometric functions, their principal values, properties, and how to solve equations involving them. By the end of this chapter, you will be able to use inverse trigonometric functions to solve advanced mathematical problems.

Key Concepts

• Inverse Trigonometric Functions: Functions that reverse the effect of trigonometric functions (e.g., sin^-1x, cos^-1x, tan^-1x, etc.).
• Principal Value Branch: The unique value of the inverse function within a specified range.
• Domain and Range: The set of input and output values for each inverse trigonometric function.

Definition and Notation

• If y = sin^-1x, then x = sin y and y ∈ [−π/2, π/2]
• If y = cos^-1x, then x = cos y and y ∈ [0, π]
• If y = tan^-1x, then x = tan y and y ∈ (−π/2, π/2)
• Similarly for cot^-1x, sec^-1x, and cosec^-1x

Principal Value Branches

• sin^-1x: x ∈ [−1, 1], principal value ∈ [−π/2, π/2]
• cos^-1x: x ∈ [−1, 1], principal value ∈ [0, π]
• tan^-1x: x ∈ ℝ, principal value ∈ (−π/2, π/2)
• cot^-1x: x ∈ ℝ, principal value ∈ (0, π)
• sec^-1x: x ∈ (−∞, −1] ∪ [1, ∞), principal value ∈ [0, π], y ≠ π/2
• cosec^-1x: x ∈ (−∞, −1] ∪ [1, ∞), principal value ∈ [−π/2, π/2], y ≠ 0

Properties and Formulas

• sin^-1(−x) = −sin^-1x
• cos^-1(−x) = π − cos^-1x
• tan^-1(−x) = −tan^-1x
• sin^-1x + cos^-1x = π/2
• tan^-1x + cot^-1x = π/2
• 2tan^-1x = sin^-1(2x/(1+x²)), for |x| < 1

Graphs of Inverse Trigonometric Functions

The graphs of inverse trigonometric functions are the reflection of the graphs of the original trigonometric functions (restricted to their principal value branches) about the line y = x.

Applications

• Solving equations involving trigonometric functions.
• Finding angles when the value of a trigonometric ratio is known.
• Used in calculus, integration, and real-life problems involving angles and distances.

Practice Questions

1. Find the principal value of sin^-1(1/2).
2. Evaluate cos^-1(−1).
3. Solve for x: tan^-1x = π/4.
4. Simplify: sin^-1x + cos^-1x.
5. If tan^-1a + tan^-1b = π/4, find the value of a + b / (1 − ab).

Challenge Yourself

• Prove that tan^-1x + tan^-1y = tan^-1((x + y)/(1 − xy)), if xy < 1.
• Draw the graph of y = sin^-1x and mark its domain and range.
• If sin^-1x = θ, express cos^-1x in terms of θ.

Did You Know?

• Inverse trigonometric functions are also called "arc functions" (e.g., arcsin x, arccos x).
• They are widely used in engineering, physics, and computer graphics.

Glossary

• Principal Value: The unique value of an inverse trigonometric function within its defined range.
• Domain: The set of all possible input values for a function.
• Range: The set of all possible output values for a function.

Answers to Practice Questions

1. π/6
2. π
3. x = 1
4. π/2
5. (a + b) / (1 − ab) = 1

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Practice using inverse trigonometric functions to solve equations and explore their properties!