AIME Trigonometry and Complex Numbers

Question 1

What is sin(30°) + cos(60°)?

Accepted Answer

1

Answer

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).

Accepted Answer

4/5

Answer

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°)?

Accepted Answer

1

Answer

tan(45°)=sin(45°)/cos(45°)=(√2/2)/(√2/2)=1.

Question 4

Find the modulus of the complex number z = 3 + 4i.

Accepted Answer

5

Answer

|z|=√(3^2+4^2)=√(9+16)=√25=5.

Question 5

Compute (2+3i)(1-i).

Accepted Answer

5+i

Answer

(2+3i)(1-i)=2-2i+3i-3i^2=2+i+3=5+i.

AIME - American Invitational Mathematics Examination Practice Test