First slide

Trigonometric equations

Question

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

Moderate

A

0
B

1
C

2
D

4

Solution

$\begin{array}{l} 2 \sin^{2} θ - \cos 2 θ = 0 \\ \Rightarrow 2 \sin^{2} θ - 1 + 2 \sin^{2} θ = 0 \\ \Rightarrow \sin θ = \pm \frac{1}{2} \\ 2 \cos^{2} θ - 3 \sin θ = 0 \\ \Rightarrow 2 \sin^{2} θ + 3 \sin θ - 2 = 0 \\ \Rightarrow \sin θ = \frac{1}{2} or - 2 \end{array}$
$∴ \sin θ = \frac{1}{2}$ (common values)
$\Rightarrow θ = \frac{π}{6}, \frac{5 π}{6}$ $(∴ θ \in [0, 2 π])$

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