Question

solve the equation Sin^
2 theeta-2costheeta+1/4=0​

Answer 1

Step-by-step explanation:

We have,

$\sin^{2} ( \theta) - 2 \cos(\theta) + \frac{1}{4} = 0$

$\implies4 \sin^{2} (\theta) - 8 \cos(\theta) + 1 = 0$

$\implies4(1 - \cos ^{2} (\theta) ) - 8 \cos(\theta) + 1 = 0$

$\implies4 - 4 \cos ^{2} (\theta) - 8 \cos(\theta) + 1 = 0$

$\implies4 \cos ^{2} (\theta) + 8 \cos(\theta) - 5= 0$

$\implies4 \cos ^{2} (\theta) + 10\cos(\theta) - 2 \cos(\theta) - 5= 0$

$\implies2\cos (\theta)(2 \cos(\theta) +5) - 1(2 \cos(\theta) + 5)= 0$

$\implies(2\cos (\theta ) - 1)(2 \cos(\theta) +5) = 0$

$\implies2\cos (\theta ) - 1 = 0 \: \: or \: \: 2 \cos(\theta) +5 = 0$

$\implies\cos (\theta ) = \frac{1}{2} \: \: or \: \: \cos(\theta) = - \frac{5}{2}$

We know, the values of cos(θ) lies between -1 to 1, so cos(θ) ≠ -5/2

So,

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

$\implies\cos (\theta ) = \cos( \frac{\pi}{3} )$

$\implies\theta = 2n\pi + \frac{\pi}{3} \: \: or \: \: 2n\pi - \frac{\pi}{3}$