First slide

Trigonometric equations

Question

The numbers of solutions of the pair of equations $2 \sin^{2} θ - \cos 2 θ = 0$ , $2 \cos^{2} θ - 3 \sin θ = 0$ in the interval $[0, 2 π]$ is

Moderate

A

zero
B

one
C

two
D

four

Solution

$2 \sin^{2} θ - \cos 2 θ = 0$
$\Rightarrow 1 - \cos 2 θ - \cos 2 θ = 0 \Rightarrow \cos 2 θ = 1 / 2$
$\Rightarrow 2 \cos^{2} θ - 1 = 1 / 2 \Rightarrow 2 \cos^{2} θ = 3 / 2$
So that from $2 \cos^{2} θ - 3 \sin θ = 0$ , we have
$\sin θ = 1 / 2 \Rightarrow θ = π / 6, 5 π / 6$ as $θ \in [0, 2 π]$ .

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