Podcast
Questions and Answers
What is the simplified form of $tan^{-1}\frac{1 - cosx}{\sqrt{1 + cosx}}$, where $0 < x < \pi$?
What is the simplified form of $tan^{-1}\frac{1 - cosx}{\sqrt{1 + cosx}}$, where $0 < x < \pi$?
Which of the following holds true for $sec^{-1}x + cosec^{-1}x$ when $|x| \ge 1$?
Which of the following holds true for $sec^{-1}x + cosec^{-1}x$ when $|x| \ge 1$?
What is the result of $2tan^{-1}\frac{1}{2} + tan^{-1}\frac{1}{7}$?
What is the result of $2tan^{-1}\frac{1}{2} + tan^{-1}\frac{1}{7}$?
For $tan^{-1}\frac{x}{\sqrt{a^2-x^2}}$, where $|x| < a$, what is its simplest form?
For $tan^{-1}\frac{x}{\sqrt{a^2-x^2}}$, where $|x| < a$, what is its simplest form?
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What does the expression $3cos^{-1}x$ equal for $-\frac{1}{2} \le x \le \frac{1}{2}$?
What does the expression $3cos^{-1}x$ equal for $-\frac{1}{2} \le x \le \frac{1}{2}$?
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Study Notes
Trigonometric Identities and Equations
- Example 7: Prove tan⁻¹x + tan⁻¹(2x/(1-x²)) = tan⁻¹(3x-x³)/(1-3x²) for |x| < 1/√3
- Solution: Substitute x = tanθ. Simplify using trigonometric identities
- Result: Left-hand side (LHS) equals right-hand side (RHS)
Trigonometric Equation Example 8
- Example 8: Find the value of cos(sec⁻¹x + cosec⁻¹x) for |x| ≥ 1
- Solution: cos(sec⁻¹x + cosec⁻¹x) = cos (π/2) = 0
- Note: This is related to the sum of inverse trigonometric functions and properties of secant and cosecant.
Trigonometric Equations (Exercise 2.2)
- Problem 1: Prove 3sin⁻¹x = sin⁻¹(3x - 4x³), where |x| ≤ 1/2
- Problem 2: Prove 3cos⁻¹x = cos⁻¹(4x³ - 3x), where |x| ≤ 1
- Problem 3: Solve tan⁻¹(1/2) + tan⁻¹(1/7) = tan⁻¹(24/7)
- Problem 4: Solve 2tan⁻¹(1/2) + tan⁻¹(1/7) = tan⁻¹(31/17)
Simplifying Trigonometric Functions
- Problem 5: Simplify tan⁻¹(√(1+x²)-1)/x, x ≠ 0
- Problem 6: Simplify tan⁻¹(1/√(x²-1)) |x| > 1
- Problem 7: Simplify tan⁻¹((1-cosx)/(1+cosx)) for 0 < x < π
- Problem 8: Simplify tan⁻¹(cosx - sinx)/(cosx + sinx) for 0 < x < π
Trigonometric Expressions involving inverse trigonometric functions
- Problem 9: Evaluate tan⁻¹(x/√(a²-x²)), where |x| < a
- Problem 10: Find the value of tan⁻¹(3a²x-x³)/(a³-3ax²) , for 0 < a, -a < x < a and √3 < x < √3
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Description
This quiz covers various examples and problems related to trigonometric identities and equations. Students will work through proofs, simplifications, and solving specific trigonometric equations involving inverse functions. It's essential for mastering the application of these concepts in mathematics.
