First slide

Trigonometric equations

Question

If $0 < x, y < 2 π$ , the number of solutions of the system of equations $\sin x \sin y = 3 / 4$ and $\cos x \cos y = 1 / 4$ is

Moderate

A

0
B

1
C

2
D

infinite

Solution

$\cos (x - y) = 1 \Rightarrow x - y = 0$ (Note $Note x - y \neq 2 π$ )
$\Rightarrow x = y$
$\cos (x + y) = - 1 / 2 \Rightarrow x + y = 120^{\circ}$ or
$240^{\circ} \Rightarrow x = y = 60^{\circ}$
or $x = y = 120^{\circ}$

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