AIME Trigonometry and Complex Numbers 2

Question 1

In a right triangle, if sin(A) = 5/13, what is cos(A)?

Accepted Answer

12/13

Answer

cos(A)=√(1-25/169)=√(144/169)=12/13.

Question 2

What is the period of f(x) = sin(3x)?

Accepted Answer

2π/3

Answer

The period of sin(kx) is 2π/k; for k=3, period=2π/3.

Question 3

Compute the conjugate of z = 7 - 2i.

Accepted Answer

7 + 2i

Answer

The complex conjugate of a+bi is a-bi; for 7-2i the conjugate is 7+2i.

Question 4

Simplify (2+i)/(1-i).

Accepted Answer

(1+3i)/2

Answer

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.

Accepted Answer

45° and 135°

Answer

sin(x)=√2/2, so x=45° and x=135° in [0°,360°).

AIME - American Invitational Mathematics Examination Practice Test