AMC 12 AMC 12 Trigonometry & Complex Numbers

Question 1

What is the value of sin(π/6)?

Accepted Answer

1/2

Answer

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?

Accepted Answer

5

Answer

The modulus is √(3² + 4²) = √(9 + 16) = √25 = 5.

Question 3

If z = 1 + i, what is z²?

Accepted Answer

2i

Answer

(1 + i)² = 1 + 2i + i² = 1 + 2i − 1 = 2i.

Question 4

Which expression equals cos(2θ)?

Accepted Answer

2cos²(θ) − 1

Answer

The double-angle identity gives cos(2θ) = 2cos²(θ) − 1.

Question 5

What is the argument (principal angle) of the complex number −1 + i?

Accepted Answer

3π/4

Answer

Since −1 + i lies in the second quadrant, its argument is π − π/4 = 3π/4.

AMC 12 - - American Mathematics Contest 12 Practice Test