To solve the equation:
[tex]\[ 4 \sin^2 \theta = 1 \][/tex]
we can follow these steps:
1. Isolate [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[ 4 \sin^2 \theta = 1 \][/tex]
Divide both sides by 4:
[tex]\[ \sin^2 \theta = \frac{1}{4} \][/tex]
2. Take the square root of both sides:
[tex]\[ \sin \theta = \pm \frac{1}{2} \][/tex]
This gives us two equations to consider:
[tex]\[ \sin \theta = \frac{1}{2} \][/tex]
[tex]\[ \sin \theta = -\frac{1}{2} \][/tex]
3. Determine the values of [tex]\(\theta\)[/tex] that satisfy each equation:
- For [tex]\(\sin \theta = \frac{1}{2}\)[/tex]:
The general solutions in radians where the sine function is [tex]\(\frac{1}{2}\)[/tex] are:
[tex]\[ \theta = \frac{\pi}{6} + 2k\pi \][/tex]
[tex]\[ \theta = \frac{5\pi}{6} + 2k\pi \][/tex]
where [tex]\(k\)[/tex] is any integer.
- For [tex]\(\sin \theta = -\frac{1}{2}\)[/tex]:
The general solutions in radians where the sine function is [tex]\(-\frac{1}{2}\)[/tex] are:
[tex]\[ \theta = \frac{7\pi}{6} + 2k\pi \][/tex]
[tex]\[ \theta = \frac{11\pi}{6} + 2k\pi \][/tex]
where [tex]\(k\)[/tex] is any integer.
Since the question does not specify a range for [tex]\(\theta\)[/tex], we often restrict [tex]\(\theta\)[/tex] to one full cycle of the sine function, typically from [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex].
4. List the solutions within [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex]:
We look for [tex]\(\theta\)[/tex] values within the standard interval [tex]\( [0, 2\pi) \)[/tex]:
- [tex]\(\theta = \frac{\pi}{6}\)[/tex]
- [tex]\(\theta = \frac{5\pi}{6}\)[/tex]
- [tex]\(\theta = \frac{7\pi}{6}\)[/tex]
- [tex]\(\theta = \frac{11\pi}{6}\)[/tex]
However, we can simplify [tex]\(\frac{11\pi}{6}\)[/tex] as combining half cycle as: [tex]\(\theta = 2\pi - \frac{\pi}{6} = \frac{7\pi}{6}\)[/tex].
Therefore, within the interval [tex]\( [0, 2\pi) \)[/tex], the solutions are:
[tex]\[
\theta = -\frac{\pi}{6}, \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}
\][/tex]
Thus, the solutions to the equation [tex]\(4 \sin^2 \theta = 1\)[/tex] are:
[tex]\[
\theta = -\frac{\pi}{6}, \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}
\][/tex]
