To solve the equation [tex]\(3 - 2 \sin x = \frac{13}{4}\)[/tex] for [tex]\(0^\circ \leq x \leq 360^\circ\)[/tex], follow these steps:
1. Isolate the trigonometric function.
Start by isolating [tex]\(\sin x\)[/tex]:
[tex]\[
3 - 2 \sin x = \frac{13}{4}
\][/tex]
Subtract 3 from both sides:
[tex]\[
-2 \sin x = \frac{13}{4} - 3
\][/tex]
Simplify the right-hand side:
[tex]\[
-2 \sin x = \frac{13}{4} - \frac{12}{4} = \frac{1}{4}
\][/tex]
2. Solve for [tex]\(\sin x\)[/tex].
Divide both sides by -2:
[tex]\[
\sin x = -\frac{1}{8}
\][/tex]
3. Determine the general solutions.
The solution to [tex]\(\sin x = -\frac{1}{8}\)[/tex] in the interval [tex]\(0^\circ \leq x \leq 360^\circ\)[/tex] involves finding the reference angle and considering the quadrants where the sine function is negative (i.e., the third and fourth quadrants).
Compute the reference angle:
[tex]\[
\theta = \arcsin\left(\frac{1}{8}\right)
\][/tex]
Approximate using a calculator:
[tex]\[
\theta \approx 7.18^\circ
\][/tex]
4. Find the specific solutions in the required interval.
Since sine is negative in the third and fourth quadrants, the angles will be:
[tex]\[
x = 180^\circ + \theta \quad \text{(Third quadrant)}
\][/tex]
[tex]\[
x = 360^\circ - \theta \quad \text{(Fourth quadrant)}
\][/tex]
Substitute [tex]\(\theta \approx 7.18^\circ\)[/tex]:
[tex]\[
x \approx 180^\circ + 7.18^\circ = 187.18^\circ
\][/tex]
[tex]\[
x \approx 360^\circ - 7.18^\circ = 352.82^\circ
\][/tex]
5. Write the final solutions.
Therefore, the solutions in the interval [tex]\(0^\circ \leq x \leq 360^\circ\)[/tex] are:
[tex]\[
x \approx 187.18^\circ \quad \text{and} \quad x \approx 352.82^\circ
\][/tex]
So, the final solutions for [tex]\( x \)[/tex] are:
[tex]\[
x \approx 187.18^\circ \quad \text{or} \quad 352.82^\circ
\][/tex]
