Answer :
To determine [tex]\(\cos 30^\circ\)[/tex], we need to use the properties of trigonometric functions on a unit circle or from a 30-60-90 special right triangle. Here is the step-by-step explanation:
1. Understand a 30-60-90 Triangle:
A 30-60-90 triangle is a special right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The side lengths have a specific ratio:
- The side opposite the 30° angle is [tex]\( \frac{1}{2} \)[/tex] of the hypotenuse.
- The side opposite the 60° angle is [tex]\(\frac{\sqrt{3}}{2}\)[/tex] of the hypotenuse.
- The hypotenuse is the longest side of the triangle.
2. Unit Circle Context:
In the unit circle, the hypotenuse (radius) is 1. Thus, for a 30° angle:
- Adjacent side to 30° (corresponds to the x-coordinate) is [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Opposite side to 30° (corresponds to the y-coordinate) is [tex]\(\frac{1}{2}\)[/tex]
3. Cosine Definition:
The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse.
[tex]\[ \cos 30^\circ = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} = \frac{\frac{\sqrt{3}}{2}}{1} = \frac{\sqrt{3}}{2} \][/tex]
4. Match with the Given Options:
Among the given options:
- [tex]\(A = \frac{1}{\sqrt{3}}\)[/tex]
- [tex]\(B = 1\)[/tex]
- [tex]\(C = \frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(D = \frac{1}{2}\)[/tex]
- [tex]\(E = \sqrt{3}\)[/tex]
- [tex]\(F = \frac{1}{\sqrt{2}}\)[/tex]
The correct answer that matches [tex]\(\cos 30^\circ\)[/tex] is option [tex]\(C\)[/tex].
So, [tex]\(\boxed{\frac{\sqrt{3}}{2}}\)[/tex].
1. Understand a 30-60-90 Triangle:
A 30-60-90 triangle is a special right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The side lengths have a specific ratio:
- The side opposite the 30° angle is [tex]\( \frac{1}{2} \)[/tex] of the hypotenuse.
- The side opposite the 60° angle is [tex]\(\frac{\sqrt{3}}{2}\)[/tex] of the hypotenuse.
- The hypotenuse is the longest side of the triangle.
2. Unit Circle Context:
In the unit circle, the hypotenuse (radius) is 1. Thus, for a 30° angle:
- Adjacent side to 30° (corresponds to the x-coordinate) is [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Opposite side to 30° (corresponds to the y-coordinate) is [tex]\(\frac{1}{2}\)[/tex]
3. Cosine Definition:
The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse.
[tex]\[ \cos 30^\circ = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} = \frac{\frac{\sqrt{3}}{2}}{1} = \frac{\sqrt{3}}{2} \][/tex]
4. Match with the Given Options:
Among the given options:
- [tex]\(A = \frac{1}{\sqrt{3}}\)[/tex]
- [tex]\(B = 1\)[/tex]
- [tex]\(C = \frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(D = \frac{1}{2}\)[/tex]
- [tex]\(E = \sqrt{3}\)[/tex]
- [tex]\(F = \frac{1}{\sqrt{2}}\)[/tex]
The correct answer that matches [tex]\(\cos 30^\circ\)[/tex] is option [tex]\(C\)[/tex].
So, [tex]\(\boxed{\frac{\sqrt{3}}{2}}\)[/tex].
