Question

Find the principal solution of the angle in the equation: 2 cos² θ = 1.

Answer 1

2 cos^2θ=1

cos^2θ=1/2

cos θ=1/√2

cos θ=cos45°

So θ=45°

Answer 2

Solution:

To find the principal solution of the angle in the equation: 2 cos² θ = 1

${cos}^{2}\theta = \frac{1}{2} \\ cos \: \theta = \sqrt{ \frac{1}{2} } \\ \\ cos \: \theta = ±\frac{1}{ \sqrt{2} }$

if

$cos \: \theta = \frac{1}{ \sqrt{2} } \\ \\ cos \: \theta = cos \: ( \frac{\pi}{4}) \\ \\ \theta = \frac{\pi}{4}$
in case of

$cos \: \theta = - \frac{1}{ \sqrt{2} } \\ \\ cos \: \theta = cos \: \frac{ - \pi}{4} \\ \\ = cos \: \frac{\pi}{4}$

since

$cos \: ( - \theta) = cos \: \theta$

so principal solution of the angle is

$\theta = \frac{\pi}{4} + 2\pi \: k$
where k is any integer