AIME Trigonometry and Complex Numbers 1 — Questions and Answers
Question 1: What is sin(30°) + cos(60°)?
- 1 (Correct answer)
- √3/2
- √2
- 1/2
Correct answer: 1
sin(30°)=1/2 and cos(60°)=1/2; sum=1.
Question 2: If cos(θ) = 3/5 and θ is in the first quadrant, find sin(θ).
- 4/5 (Correct answer)
- 3/5
- 5/4
- 5/3
Correct answer: 4/5
sin^2(θ)+cos^2(θ)=1 gives sin^2(θ)=1-9/25=16/25, so sin(θ)=4/5.
Question 3: What is the value of tan(45°)?
- 1 (Correct answer)
- 0
- √3
- 1/√3
Correct answer: 1
tan(45°)=sin(45°)/cos(45°)=(√2/2)/(√2/2)=1.
Question 4: Find the modulus of the complex number z = 3 + 4i.
- 5 (Correct answer)
- 7
- √7
- 1
Correct answer: 5
|z|=√(3^2+4^2)=√(9+16)=√25=5.
Question 5: Compute (2+3i)(1-i).
- 5+i (Correct answer)
- 2-i
- 5-i
- -1+i
Correct answer: 5+i
(2+3i)(1-i)=2-2i+3i-3i^2=2+i+3=5+i.
Question 6: What is i^10 (where i = √(-1))?
- -1 (Correct answer)
- 1
- i
- -i
Correct answer: -1
i^10=(i^4)^2·i^2=1·(-1)=-1.
AIME Trigonometry and Complex Numbers 1
What is sin(30°) + cos(60°)?