Given the trigonometric equation [tex]\(\operatorname{Arccos} \frac{\sqrt{3}}{2} = \beta\)[/tex], we need to determine the angle [tex]\(\beta\)[/tex] whose cosine is [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
To solve for [tex]\(\beta\)[/tex]:
1. Understand the Problem:
- The function [tex]\(\operatorname{Arccos}(x)\)[/tex] gives the angle whose cosine is [tex]\(x\)[/tex].
- Therefore, [tex]\(\operatorname{Arccos} \frac{\sqrt{3}}{2} = \beta\)[/tex] means that [tex]\(\cos(\beta) = \frac{\sqrt{3}}{2}\)[/tex].
2. Recall Standard Angles:
- We need to recall which standard angle has a cosine value of [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
- From standard angles in trigonometry, we know that [tex]\(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)[/tex].
3. Determine Beta in Degrees:
- Since [tex]\(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)[/tex], we conclude that [tex]\(\beta = 30^\circ\)[/tex].
4. Convert Beta to Radians:
- Angles can also be represented in radians. Knowing that [tex]\(180^\circ = \pi\)[/tex] radians, we convert [tex]\(30^\circ\)[/tex] to radians:
[tex]\[
\beta = 30^\circ \times \frac{\pi \text{ radians}}{180^\circ} = \frac{\pi}{6} \text{ radians}.
\][/tex]
5. Provide the Detailed Solution:
- The angle [tex]\(\beta\)[/tex] in radians is [tex]\(\frac{\pi}{6}\)[/tex] radians.
- The angle [tex]\(\beta\)[/tex] in degrees is [tex]\(30^\circ\)[/tex].
Thus, [tex]\(\operatorname{Arccos} \frac{\sqrt{3}}{2}\)[/tex] is:
- [tex]\(\beta \approx 0.5236 \text{ radians}\)[/tex]
- [tex]\(\beta \approx 30.0^\circ\)[/tex]
Therefore, the exact values are [tex]\(\beta = \frac{\pi}{6}\)[/tex] radians and [tex]\(\beta = 30^\circ\)[/tex]. The approximate numerical results confirm these values:
[tex]\[
\boxed{0.5236 \text{ radians} \text{ or } 30.0^\circ}
\][/tex]
