To solve the equation [tex]\(4 + 2 \sin x = 14 - 8 \sin x\)[/tex] for [tex]\(0^{\circ} \leq x \leq 180^{\circ}\)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[4 + 2 \sin x = 14 - 8 \sin x\][/tex]
2. Isolate the sine term on one side of the equation.
Subtract [tex]\(2 \sin x\)[/tex] from both sides of the equation:
[tex]\[4 = 14 - 10 \sin x\][/tex]
3. Move the constant term to the other side:
Subtract 14 from both sides:
[tex]\[4 - 14 = -10 \sin x\][/tex]
Simplifying this:
[tex]\[-10 = -10 \sin x\][/tex]
4. Solve for [tex]\(\sin x\)[/tex]:
Divide both sides by -10:
[tex]\[1 = \sin x\][/tex]
5. Determine the angle [tex]\(x\)[/tex] where the sine of [tex]\(x\)[/tex] is 1.
Recall that the sine function equals 1 at [tex]\(90^{\circ}\)[/tex].
Since the given range is [tex]\(0^{\circ} \leq x \leq 180^{\circ}\)[/tex], [tex]\(x = 90^{\circ}\)[/tex] is the solution within this range.
6. Conclusion:
The solution to the equation [tex]\(4 + 2 \sin x = 14 - 8 \sin x\)[/tex] within the interval [tex]\([0^{\circ}, 180^{\circ}]\)[/tex] is:
[tex]\[\boxed{90^{\circ}}\][/tex]
