MathDB

Double Trigonometric Ratios

Source: 2012 AIME II Problem 9

March 29, 2012

ratiotrigonometryfunctionquadraticsnumber theoryrelatively primeAMC

Problem Statement

Let

x

and

y

be real numbers such that

\frac{\sin{x}}{\sin{y}} = 3

and

\frac{\cos{x}}{\cos{y}} = \frac{1}{2}

. The value of

\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}

can be expressed in the form

\frac{p}{q}

, where

p

and

q

are relatively prime positive integers. Find

p + q