AMC 12 AMC 12 Trigonometry & Complex Numbers 1 — Questions and Answers
Question 1: What is the value of sin(π/6)?
- 1/2 (Correct answer)
- √3/2
- √2/2
- 1
Correct answer: 1/2
sin(π/6) = sin(30°) = 1/2 by the standard unit circle values.
Question 2: What is the modulus of the complex number 3 + 4i?
- 3
- 4
- 5 (Correct answer)
- 7
Correct answer: 5
The modulus is √(3² + 4²) = √(9 + 16) = √25 = 5.
Question 3: If z = 1 + i, what is z²?
- 2
- 2i (Correct answer)
- 1 + 2i
- -1 + 2i
Correct answer: 2i
(1 + i)² = 1 + 2i + i² = 1 + 2i − 1 = 2i.
Question 4: Which expression equals cos(2θ)?
- 2cos(θ) − 1
- 2cos²(θ) − 1 (Correct answer)
- 1 − 2cos(θ)
- cos(θ) − sin(θ)
Correct answer: 2cos²(θ) − 1
The double-angle identity gives cos(2θ) = 2cos²(θ) − 1.
Question 5: What is the argument (principal angle) of the complex number −1 + i?
- π/4
- 3π/4 (Correct answer)
- −π/4
- π/6
Correct answer: 3π/4
Since −1 + i lies in the second quadrant, its argument is π − π/4 = 3π/4.
Question 6: What is the value of sin²(θ) + cos²(θ) for all θ?
- 0
- 2
- 1 (Correct answer)
- sin(2θ)
Correct answer: 1
The Pythagorean identity states sin²(θ) + cos²(θ) = 1 for all real θ.
AMC 12 AMC 12 Trigonometry & Complex Numbers 1
What is the value of sin(π/6)?