To find the value of [tex]\(\theta\)[/tex] given the equation [tex]\(6 \sin \theta = 5\)[/tex], we will follow these steps:
1. Isolate [tex]\(\sin \theta\)[/tex]:
We need to isolate [tex]\(\sin \theta\)[/tex] by dividing both sides of the equation by 6:
[tex]\[
\sin \theta = \frac{5}{6}
\][/tex]
2. Find [tex]\(\theta\)[/tex] in radians:
To find the angle [tex]\(\theta\)[/tex], we use the inverse sine function (also known as arcsin). This function will give us [tex]\(\theta\)[/tex] in radians.
[tex]\[
\theta = \sin^{-1}\left(\frac{5}{6}\right)
\][/tex]
Using this inverse sine function, we get:
[tex]\[
\theta \approx 0.9851 \text{ radians}
\][/tex]
3. Convert [tex]\(\theta\)[/tex] from radians to degrees:
To convert the angle from radians to degrees, we use the conversion factor [tex]\(\frac{180^\circ}{\pi}\)[/tex]:
[tex]\[
\theta \approx 0.9851 \times \frac{180^\circ}{\pi}
\][/tex]
By performing this calculation, we find:
[tex]\[
\theta \approx 56.4^\circ
\][/tex]
4. Provide the answer to 1 decimal place:
The final value of [tex]\(\theta\)[/tex] is:
[tex]\[
\theta \approx 56.4^\circ
\][/tex]
Therefore, the value of [tex]\(\theta\)[/tex], correct to one decimal place, is [tex]\(56.4^\circ\)[/tex].
