Trigonometry Inverse Functions 3 â Questions and Answers
Question 1: Evaluate arccos(-1/2).
- Ď/3
- 2Ď/3 (Correct answer)
- 5Ď/6
- Ď/6
Correct answer: 2Ď/3
cos(2Ď/3) = -1/2 and 2Ď/3 is in [0, Ď], so arccos(-1/2) = 2Ď/3.
Question 2: If arctan(x) = Ď/4, what is x?
- â2/2
- 1 (Correct answer)
- â3
- 1/2
Correct answer: 1
tan(Ď/4) = 1, so x = 1.
Question 3: What is the domain of arctan(x)?
- [-1, 1]
- [0, Ď]
- (-â, â) (Correct answer)
- (-Ď/2, Ď/2)
Correct answer: (-â, â)
The tangent function is defined for all real inputs, so arctan(x) accepts all real x.
Question 4: Evaluate arcsin(-â2/2).
- Ď/4
- -Ď/4 (Correct answer)
- 3Ď/4
- -3Ď/4
Correct answer: -Ď/4
sin(-Ď/4) = -â2/2 and -Ď/4 is in [-Ď/2, Ď/2], so the answer is -Ď/4.
Question 5: Which expression equals tan(arcsin(x)) for |x| < 1?
- x/â(1-x²) (Correct answer)
- â(1-x²)/x
- x/â(1+x²)
- 1/â(1-x²)
Correct answer: x/â(1-x²)
If sin θ = x, then cos θ = â(1-x²), so tan θ = x/â(1-x²).
Question 6: What is the value of arctan(-1)?
- -Ď/4 (Correct answer)
- Ď/4
- -3Ď/4
- 3Ď/4
Correct answer: -Ď/4
tan(-Ď/4) = -1 and -Ď/4 is in (-Ď/2, Ď/2), so arctan(-1) = -Ď/4.
Question 7: Which equation correctly represents a property of inverse trig functions?
- sin(arcsin(2)) = 2
- arcsin(sin(3Ď/2)) = 3Ď/2
- arcsin(sin(Ď/6)) = Ď/6 (Correct answer)
- cos(arcsin(x)) = x
Correct answer: arcsin(sin(Ď/6)) = Ď/6
Ď/6 lies in [-Ď/2, Ď/2], so arcsin(sin(Ď/6)) = Ď/6 holds; the others involve values outside the principal range.
