To solve the equation [tex]\(4 + 2 \sin x = 14 - 8 \sin x\)[/tex] for [tex]\(0^{\circ} \leq x \leq 180^{\circ}\)[/tex], we can follow these steps:
1. Combine like terms to isolate [tex]\(\sin x\)[/tex]:
[tex]\[
4 + 2 \sin x = 14 - 8 \sin x
\][/tex]
Move [tex]\(8 \sin x\)[/tex] to the left side of the equation by adding [tex]\(8 \sin x\)[/tex] to both sides:
[tex]\[
4 + 2 \sin x + 8 \sin x = 14
\][/tex]
Simplify:
[tex]\[
4 + 10 \sin x = 14
\][/tex]
2. Isolate [tex]\(\sin x\)[/tex] on one side of the equation:
Subtract 4 from both sides:
[tex]\[
10 \sin x = 14 - 4
\][/tex]
Simplify:
[tex]\[
10 \sin x = 10
\][/tex]
Divide both sides by 10:
[tex]\[
\sin x = 1
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
We need to find the value(s) of [tex]\(x\)[/tex] such that [tex]\(\sin x = 1\)[/tex] within the range [tex]\(0^{\circ} \leq x \leq 180^{\circ}\)[/tex].
Recall that [tex]\(\sin x = 1\)[/tex] at [tex]\(x = 90^{\circ}\)[/tex].
Thus, the solution to the equation [tex]\(4 + 2 \sin x = 14 - 8 \sin x\)[/tex] within the specified range is:
[tex]\[
x = 90^{\circ}
\][/tex]
