To determine the solutions to the equation [tex]\(\cos\left(\frac{x}{2}\right) - \sin(x) = 0\)[/tex] for [tex]\(0^\circ \le x < 360^\circ\)[/tex], let's proceed step-by-step:
1. Rewrite the Equation:
We start with the given trigonometric equation:
[tex]\[
\cos\left(\frac{x}{2}\right) - \sin(x) = 0
\][/tex]
This can be rearranged to:
[tex]\[
\cos\left(\frac{x}{2}\right) = \sin(x)
\][/tex]
2. Possible Solution Sets:
We'll consider different possible sets of angles within the given range ([tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex]).
3. Check the Given Options:
Our task is to determine which set of angles satisfies the equation.
a. Check [tex]\(\left\{0^\circ, 60^\circ, 300^\circ\right\}\)[/tex]:
- For [tex]\(x = 0^\circ\)[/tex]:
[tex]\[
\cos\left(\frac{0^\circ}{2}\right) = \cos(0^\circ) = 1
\][/tex]
[tex]\[
\sin(0^\circ) = 0
\][/tex]
[tex]\[
\cos(0^\circ) \neq \sin(0^\circ)
\][/tex]
So, [tex]\(0^\circ\)[/tex] does not satisfy the equation.
b. Check [tex]\(\left\{0^\circ, 120^\circ, 240^\circ\right\}\)[/tex]:
- We already saw that [tex]\(0^\circ\)[/tex] doesn't satisfy the equation, so this set also does not satisfy the equation.
c. Check [tex]\(\left\{60^\circ, 180^\circ, 300^\circ\right\}\)[/tex]:
- For [tex]\(x = 60^\circ\)[/tex]:
[tex]\[
\cos\left(\frac{60^\circ}{2}\right) = \cos(30^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\cos(30^\circ) = \sin(60^\circ)
\][/tex]
So, [tex]\(60^\circ\)[/tex] satisfies the equation.
- For [tex]\(x = 180^\circ\)[/tex]:
[tex]\[
\cos\left(\frac{180^\circ}{2}\right) = \cos(90^\circ) = 0
\][/tex]
[tex]\[
\sin(180^\circ) = 0
\][/tex]
[tex]\[
\cos(90^\circ) = \sin(180^\circ)
\][/tex]
So, [tex]\(180^\circ\)[/tex] satisfies the equation.
- For [tex]\(x = 300^\circ\)[/tex]:
[tex]\[
\cos\left(\frac{300^\circ}{2}\right) = \cos(150^\circ) = -\frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\sin(300^\circ) = -\frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\cos(150^\circ) = \sin(300^\circ)
\][/tex]
So, [tex]\(300^\circ\)[/tex] satisfies the equation.
d. Check [tex]\(\left\{120^\circ, 180^\circ, 240^\circ\right\}\)[/tex]:
- For [tex]\(x = 120^\circ\)[/tex]:
[tex]\[
\cos\left(\frac{120^\circ}{2}\right) = \cos(60^\circ) = \frac{1}{2}
\][/tex]
[tex]\[
\sin(120^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\cos(60^\circ) \neq \sin(120^\circ)
\][/tex]
So, [tex]\(120^\circ\)[/tex] does not satisfy the equation.
Since [tex]\(\left\{60^\circ, 180^\circ, 300^\circ\right\}\)[/tex] is the set of angles that satisfies the equation, the correct answer is:
[tex]\(\boxed{\{60^\circ, 180^\circ, 300^\circ\}}\)[/tex]
