AMC 12 AMC 12 Trigonometry & Complex Numbers 2

Question 1

What is the product (2 + 3i)(2 − 3i)?

Accepted Answer

13

Answer

(2 + 3i)(2 − 3i) = 4 − 9i² = 4 + 9 = 13.

Question 2

In triangle ABC, angle A = 30°, b = 6, c = 4. What is the area of the triangle?

Accepted Answer

6

Answer

Area = (1/2)bc·sin(A) = (1/2)(6)(4)(1/2) = 6.

Question 3

What is tan(π/4)?

Accepted Answer

1

Answer

tan(π/4) = sin(π/4)/cos(π/4) = (√2/2)/(√2/2) = 1.

Question 4

What is the real part of (1 + 2i)/(1 − i)?

Accepted Answer

−1/2

Answer

Multiplying by the conjugate: (1+2i)(1+i)/2 = (−1+3i)/2, giving real part −1/2.

Question 5

Which expression correctly states the sine angle-addition formula?

Accepted Answer

sin A·cos B + cos A·sin B

Answer

The angle-addition identity is sin(A + B) = sin A·cos B + cos A·sin B.

AMC 12 - - American Mathematics Contest 12 Practice Test