Solve each equation on the interval 0 greater than or equal to theta or less than 2pie.
12) 4 (cos^2) theta - 3 = 0
Please be descriptive I have having difficulties with these types. TY
Just "unwrap" it like you would any other equation.
$\displaystyle 4\cos^2{\theta} - 3 = 0$ Add 3 to both sides.
$\displaystyle 4\cos^2{\theta} = 3$ Divide by 4.
$\displaystyle \cos^2{\theta} = \frac{3}{4}$. Take the square root.
$\displaystyle \cos{\theta} = \pm\frac{\sqrt{3}}{2}$.
So what values of $\displaystyle \theta$ make $\displaystyle \cos{\theta} = \frac{\sqrt{3}}{2}$ or $\displaystyle -\frac{\sqrt{3}}{2}$.
Since the answer is both positive and negative, there are answers in all quadrants. Let's start with the focus angle.
We know $\displaystyle \cos{\frac{\pi}{6}} = \frac{\sqrt{3}}{2}$
So $\displaystyle \frac{\pi}{6}$ is the focus angle.
In the domain $\displaystyle 0 \leq x \leq 2\pi$ the solutions are...
$\displaystyle x = \left\{\frac{\pi}{6}, \pi - \frac{\pi}{6}, \pi + \frac{\pi}{6}, 2\pi - \frac{\pi}{6}\right\}$
$\displaystyle = \left\{\frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}\right\}$.
Does that make sense?