Let's solve the equation [tex]\(\cos(3x + 180^\circ) = \frac{\sqrt{3}}{2}\)[/tex] for [tex]\(0^\circ \leq x \leq 180^\circ\)[/tex].
The cosine function [tex]\(\cos(\theta)\)[/tex] equals [tex]\(\frac{\sqrt{3}}{2}\)[/tex] at specific angles. These angles can be determined from the unit circle:
[tex]\[
\cos(\theta) = \frac{\sqrt{3}}{2} \implies \theta = 30^\circ + 360^\circ n \text{ or } \theta = 330^\circ + 360^\circ n \text{ for integer } n
\][/tex]
Replace [tex]\(\theta\)[/tex] with [tex]\(3x + 180^\circ\)[/tex] and solve for [tex]\(x\)[/tex].
1. Solving [tex]\(3x + 180^\circ = 30^\circ + 360^\circ n\)[/tex]:
[tex]\[
3x + 180^\circ = 30^\circ + 360^\circ n
\][/tex]
Rearrange to isolate [tex]\(x\)[/tex]:
[tex]\[
3x = 30^\circ - 180^\circ + 360^\circ n
\][/tex]
[tex]\[
3x = -150^\circ + 360^\circ n
\][/tex]
[tex]\[
x = \frac{-150^\circ + 360^\circ n}{3}
\][/tex]
We need to find [tex]\(x\)[/tex] within the range [tex]\(0^\circ \leq x \leq 180^\circ\)[/tex]. Therefore:
For [tex]\(n = -1\)[/tex]:
[tex]\[
x = \frac{-150^\circ + 360^\circ \cdot (-1)}{3} = \frac{-150^\circ - 360^\circ}{3} = \frac{-510^\circ}{3} = -170^\circ \quad (\text{out of range})
\][/tex]
For [tex]\(n = 0\)[/tex]:
[tex]\[
x = \frac{-150^\circ + 360^\circ \cdot 0}{3} = \frac{-150^\circ}{3} = -50^\circ \quad (\text{out of range})
\][/tex]
For [tex]\(n = 1\)[/tex]:
[tex]\[
x = \frac{-150^\circ + 360^\circ \cdot 1}{3} = \frac{-150^\circ + 360^\circ}{3} = \frac{210^\circ}{3} = 70^\circ \quad (\text{valid})
\][/tex]
2. Solving [tex]\(3x + 180^\circ = 330^\circ + 360^\circ n\)[/tex]:
[tex]\[
3x + 180^\circ = 330^\circ + 360^\circ n
\][/tex]
Rearrange to isolate [tex]\(x\)[/tex]:
[tex]\[
3x = 330^\circ - 180^\circ + 360^\circ n
\][/tex]
[tex]\[
3x = 150^\circ + 360^\circ n
\][/tex]
[tex]\[
x = \frac{150^\circ + 360^\circ n}{3}
\][/tex]
We need to find [tex]\(x\)[/tex] within the range [tex]\(0^\circ \leq x \leq 180^\circ\)[/tex]. Therefore:
For [tex]\(n = 0\)[/tex]:
[tex]\[
x = \frac{150^\circ + 360^\circ \cdot 0}{3} = \frac{150^\circ}{3} = 50^\circ \quad (\text{valid})
\][/tex]
For [tex]\(n = 1\)[/tex]:
[tex]\[
x = \frac{150^\circ + 360^\circ \cdot 1}{3} = \frac{150^\circ + 360^\circ}{3} = \frac{510^\circ}{3} = 170^\circ \quad (\text{valid})
\][/tex]
Combining all obtained results, [tex]\(x\)[/tex] values within the given range are:
[tex]\[
\boxed{70^\circ, 50^\circ, 170^\circ}
\][/tex]
