Let's solve the given equation [tex]\(2 \sin \theta = \sqrt{3}\)[/tex] step-by-step.
1. Isolate [tex]\(\sin \theta\)[/tex]:
[tex]\[
\sin \theta = \frac{\sqrt{3}}{2}
\][/tex]
2. Determine the angle [tex]\(\theta\)[/tex]:
To find the angle [tex]\(\theta\)[/tex] that satisfies [tex]\(\sin \theta = \frac{\sqrt{3}}{2}\)[/tex], we use the arcsin function, which is the inverse of the sine function. The sine of an angle returns the ratio of the opposite side to the hypotenuse in a right-angled triangle, and we need to find the angle whose sine is [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
[tex]\[
\theta = \arcsin \left( \frac{\sqrt{3}}{2} \right)
\][/tex]
3. Evaluate [tex]\(\arcsin \left( \frac{\sqrt{3}}{2} \right)\)[/tex]:
By calculating or recognizing this value, we find that:
[tex]\[
\theta \approx 1.0471975511965976 \text{ radians}
\][/tex]
4. Convert the angle from radians to degrees:
Many times, it is useful or required to express the angle in degrees rather than radians. We convert radians to degrees using the following conversion factor: [tex]\(1 \text{ radian} = 180/\pi \text{ degrees}\)[/tex].
[tex]\[
\theta_{\text{degrees}} = 1.0471975511965976 \times \left( \frac{180}{\pi} \right)
\][/tex]
Simplifying this, we get:
[tex]\[
\theta_{\text{degrees}} \approx 59.99999999999999 \text{ degrees}
\][/tex]
5. Conclusion:
Hence, the angle [tex]\(\theta\)[/tex] that satisfies the equation [tex]\(2 \sin \theta = \sqrt{3}\)[/tex] is approximately [tex]\(1.0471975511965976\)[/tex] radians or [tex]\(59.99999999999999\)[/tex] degrees.
6. General solution:
The general solutions for [tex]\(\theta\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] that satisfy the equation are:
[tex]\[
\theta = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad \theta = \pi - \frac{\pi}{3} + 2k\pi
\][/tex]
Simplifying, we get:
[tex]\[
\theta = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad \theta = \frac{2\pi}{3} + 2k\pi
\][/tex]
Here, [tex]\(k\)[/tex] is any integer.
To summarize:
- The primary solution for [tex]\(\theta\)[/tex] is approximately [tex]\(1.0471975511965976\)[/tex] radians or [tex]\(59.99999999999999\)[/tex] degrees.
- The general solutions are [tex]\(\theta = \frac{\pi}{3} + 2k\pi\)[/tex] and [tex]\(\theta = \frac{2\pi}{3} + 2k\pi\)[/tex] for any integer [tex]\(k\)[/tex].
