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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 $0 < x < \pi$?
- $\frac{\pi}{4}$
- $\frac{x}{2}$ (correct)
- $\frac{1}{2} \tan^{-1} x$
- $\frac{\pi}{2} - \frac{x}{2}$
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 $|x| \ge 1$?
- $\frac{3\pi}{4}$
- $\frac{\pi}{3}$
- $\frac{\pi}{2}$ (correct)
- $\frac{\pi}{4}$
What is the result of $2tan^{-1}\frac{1}{2} + tan^{-1}\frac{1}{7}$?
What is the result of $2tan^{-1}\frac{1}{2} + tan^{-1}\frac{1}{7}$?
- $tan^{-1}\frac{31}{17}$ (correct)
- $tan^{-1}\frac{31}{7}$
- $tan^{-1}\frac{1}{31}$
- $tan^{-1}\frac{17}{31}$
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?
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 $-\frac{1}{2} \le x \le \frac{1}{2}$?
Flashcards
Inverse Trigonometric Functions
Inverse Trigonometric Functions
Functions that return the angle for a given trigonometric ratio. Examples include arcsin(x), arccos(x), arctan(x), etc.
tan⁻¹(x) + tan⁻¹(2x/(1-x²))
tan⁻¹(x) + tan⁻¹(2x/(1-x²))
Equals tan⁻¹(3x-x³/1-3x²), for |x| < 1/√3
cos(sec⁻¹(x) + cosec⁻¹(x))
cos(sec⁻¹(x) + cosec⁻¹(x))
Equals 0, when |x| ≥ 1
3sin⁻¹(x) = sin⁻¹(3x-4x²)
3sin⁻¹(x) = sin⁻¹(3x-4x²)
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tan⁻¹(1/2) + tan⁻¹(1/7)
tan⁻¹(1/2) + tan⁻¹(1/7)
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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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