To solve the equation [tex]\(2 \cos \theta - 1 = 0\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex], follow these steps:
1. Isolate the cosine function:
Start by solving the equation for [tex]\(\cos \theta\)[/tex].
[tex]\[
2 \cos \theta - 1 = 0
\][/tex]
Add 1 to both sides of the equation:
[tex]\[
2 \cos \theta = 1
\][/tex]
Divide both sides by 2:
[tex]\[
\cos \theta = \frac{1}{2}
\][/tex]
2. Identify the general solutions for [tex]\(\theta\)[/tex]:
The cosine function equals [tex]\(\frac{1}{2}\)[/tex] at specific points. We know from trigonometric identities that:
[tex]\[
\cos \theta = \frac{1}{2} \quad \text{at} \quad \theta = \frac{\pi}{3} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{3} + 2k\pi
\][/tex]
where [tex]\(k\)[/tex] is any integer.
3. Filter solutions within the interval [tex]\([0, 2\pi)\)[/tex]:
We are looking for solutions within the interval [tex]\([0, 2\pi)\)[/tex]. Let's list and check them:
- [tex]\(\theta = \frac{\pi}{3}\)[/tex]
- [tex]\(\theta = \frac{5\pi}{3}\)[/tex]
Both [tex]\(\frac{\pi}{3}\)[/tex] and [tex]\(\frac{5\pi}{3}\)[/tex] lie within the interval [tex]\([0, 2\pi)\)[/tex].
4. Write the solutions in terms of [tex]\(\pi\)[/tex]:
The solutions are:
[tex]\[
\theta = \frac{\pi}{3}, \frac{5\pi}{3}
\][/tex]
So, the solutions to the equation [tex]\(2 \cos \theta - 1 = 0\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{3}, \frac{5\pi}{3}
\][/tex]
