Question

2sin² θ -2cosθ = 1/2

Answer 1

Step-by-step explanation:

We have,

$2 \sin^{2} ( \theta) - 2 \cos( \theta) = \dfrac{1}{2}$

$\implies4 \sin^{2} ( \theta) - 4\cos( \theta) = 1$

$\implies4 - 4\cos^{2} ( \theta) - 4\cos( \theta) = 1$

$\implies3 - 4\cos^{2} ( \theta) - 4\cos( \theta) = 0$

$\implies 4\cos^{2} ( \theta) + 4\cos( \theta) - 3 = 0$

$\implies 4\cos^{2} ( \theta) + 6\cos( \theta) - 2 \cos( \theta) - 3 = 0$

$\implies 2 \cos ( \theta) \big( 2\cos( \theta) + 3 \big) - 1 \big(2 \cos( \theta) + 3 \big)= 0$

$\implies \big( 2\cos( \theta) + 3 \big) \big(2 \cos( \theta) - 1\big)= 0$

$\implies 2\cos( \theta) + 3 = 0 \: \: \: \: or \: \: \: \: 2 \cos( \theta) - 1= 0$

$\implies \cos( \theta) = - \dfrac{ 3}{2} \: \: \: \: or \: \: \: \: \cos( \theta) = \dfrac{1}{2}$

$\tt{since \: \: \: cos (\theta) \in [ - 1,1]}$

So,

$\implies \cos( \theta) = \dfrac{1}{2}$

$\implies \cos( \theta) = \cos \left(\dfrac{\pi}{3} \right)$

$\implies \theta = 2n \pi \pm\dfrac{\pi}{3}$

$\implies \theta = 2n \pi \pm\dfrac{\pi}{3} \: \: \: \: \: \: \: \: \: \: , \forall \: n \in \mathbb{Z}$