Answer :
To calculate [tex]\(\cos 30^\circ\)[/tex], we first need to recognize that 30 degrees is one of the standard angles in trigonometry. The trigonometric functions for these standard angles are often derived from special triangles or the unit circle.
For [tex]\(\cos 30^\circ\)[/tex], we can use the properties of a 30-60-90 right triangle. In such a triangle, the sides are in the ratio 1:[tex]\(\sqrt{3}\)[/tex]:2.
To find [tex]\(\cos 30^\circ\)[/tex], we need to identify the adjacent side and the hypotenuse in a 30-60-90 triangle.
- The hypotenuse is the longest side, opposite the right angle, and in a 30-60-90 triangle, it is twice the length of the shorter leg, which is opposite the 30° angle.
- The adjacent side is the side next to the 30° angle, which is the longer leg of the triangle and has a length of [tex]\(\sqrt{3}\)[/tex].
The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. Thus,
[tex]\[ \cos 30^\circ = \frac{\text{adjacent side}}{\text{hypotenuse}} \][/tex]
In a 30-60-90 triangle:
[tex]\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \][/tex]
Therefore, the correct answer is:
E. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
For [tex]\(\cos 30^\circ\)[/tex], we can use the properties of a 30-60-90 right triangle. In such a triangle, the sides are in the ratio 1:[tex]\(\sqrt{3}\)[/tex]:2.
To find [tex]\(\cos 30^\circ\)[/tex], we need to identify the adjacent side and the hypotenuse in a 30-60-90 triangle.
- The hypotenuse is the longest side, opposite the right angle, and in a 30-60-90 triangle, it is twice the length of the shorter leg, which is opposite the 30° angle.
- The adjacent side is the side next to the 30° angle, which is the longer leg of the triangle and has a length of [tex]\(\sqrt{3}\)[/tex].
The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. Thus,
[tex]\[ \cos 30^\circ = \frac{\text{adjacent side}}{\text{hypotenuse}} \][/tex]
In a 30-60-90 triangle:
[tex]\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \][/tex]
Therefore, the correct answer is:
E. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
