First slide

Trigonometric equations

Question

If $6 \cos 2 θ + 2 \cos^{2} (θ / 2) + 2 \sin^{2} θ = 0$ where $- π < 0 < π$ , then $θ$ is equal to

Moderate

A

$π / 3$
B

$π / 3, \cos^{- 1} (3 / 5)$
C

$\cos^{- 1} (3 / 5)$
D

$π / 3, π - \cos^{- 1} (3 / 5)$

Solution

The given equation can be written as
$6 (2 \cos^{2} θ - 1) + (1 + \cos θ) + 2 (1 - \cos^{2} θ) = 0$
$\Rightarrow 10 \cos^{2} θ + \cos θ - 3 = 0$
$\Rightarrow (5 \cos θ + 3) (2 \cos θ - 1) = 0$
$\Rightarrow \cos θ = \frac{1}{2}$ or $\Rightarrow \cos θ = \frac{- 3}{5}$
$\Rightarrow θ = \frac{π}{3}$ or $θ = π - \cos^{- 1} (\frac{3}{5})$ as $- π < θ < π$
$\Rightarrow θ = \frac{π}{3}, π - \cos^{- 1} (\frac{3}{5})$

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