Satisfying two equations in trigonometry

Question

Suppose that x and y satisfy the equations

3 sin x + 4 cos y = 5

4 sin y + 3 cos x = 2

What is the value of sin(x + y)?

Answer 1

Oooh. This is a fun one. My advice to you is to take each of the two equations and square them. For example, if you square the first, you get:

9 sin^2(x) + 24sin(x)cos(y) + 16cos^2y = 25

Why? Because if you square them both, then add the two squared equations together you'll be able to use the relations:

sin^2(x) + cos^2(x) = 1

sin(x + y) = sin(x)cos(y) + sin(y)cos(x)

to solve for sin(x + y).

If you get an answer outside the range of -1 <= sin(x + y) <= 1, you've done something wrong.