AIME Trigonometry and Complex Numbers 2 — Questions and Answers
Question 1: In a right triangle, if sin(A) = 5/13, what is cos(A)?
- 12/13 (Correct answer)
- 5/12
- 13/12
- 13/5
Correct answer: 12/13
cos(A)=√(1-25/169)=√(144/169)=12/13.
Question 2: What is the period of f(x) = sin(3x)?
- 2π/3 (Correct answer)
- 2π
- 3π
- 6π
Correct answer: 2π/3
The period of sin(kx) is 2π/k; for k=3, period=2π/3.
Question 3: Compute the conjugate of z = 7 - 2i.
- 7 + 2i (Correct answer)
- 7 - 2i
- -7 + 2i
- -7 - 2i
Correct answer: 7 + 2i
The complex conjugate of a+bi is a-bi; for 7-2i the conjugate is 7+2i.
Question 4: Simplify (2+i)/(1-i).
- (1+3i)/2 (Correct answer)
- (3+i)/2
- (1-3i)/2
- (3-i)/2
Correct answer: (1+3i)/2
Multiply by (1+i)/(1+i): (2+i)(1+i)/((1-i)(1+i))=(2+3i+i^2)/2=(1+3i)/2.
Question 5: Find all x in [0°, 360°) satisfying 2sin(x) = √2.
- 45° and 135° (Correct answer)
- 30° and 150°
- 45° and 225°
- 60° and 120°
Correct answer: 45° and 135°
sin(x)=√2/2, so x=45° and x=135° in [0°,360°).
Question 6: If z = 1 + i, what is z^2?
- 2i (Correct answer)
- 2
- -2i
- 1+2i
Correct answer: 2i
(1+i)^2=1+2i+i^2=1+2i-1=2i.
AIME Trigonometry and Complex Numbers 2
In a right triangle, if sin(A) = 5/13, what is cos(A)?