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Determine the value of [tex]\(\beta\)[/tex].

[tex]\[\operatorname{Arccos} \frac{\sqrt{3}}{2}=\beta\][/tex]

A. [tex]\(45^{\circ}, 45^{\circ}, 90^{\circ}; 1,1, \sqrt{2}\)[/tex]

B. [tex]\(30^{\circ}, 60^{\circ}, 90^{\circ}; 1, 1, \sqrt{2}\)[/tex]

C. [tex]\(45^{\circ}, 45^{\circ}, 90^{\circ}; 1, \sqrt{3}, 2\)[/tex]

D. [tex]\(30^{\circ}, 60^{\circ}, 90^{\circ}; 1, \sqrt{3}, 2\)[/tex]



Answer :

GinnyAnswer
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]

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