To find the value of [tex]\(\cos 30^\circ\)[/tex], let's follow a clear step-by-step approach:
1. Understand the angle given: The angle provided is [tex]\(30^\circ\)[/tex].
2. Recall the fundamental trigonometric values: For certain key angles such as [tex]\(30^\circ\)[/tex], [tex]\(45^\circ\)[/tex], and [tex]\(60^\circ\)[/tex], trigonometric values are often memorized or derived from special triangles (like the 30-60-90 triangle or 45-45-90 triangle).
3. Use the 30-60-90 triangle:
- In a 30-60-90 triangle, the sides are in the ratio of [tex]\(1 : \sqrt{3} : 2\)[/tex].
- For an angle of [tex]\(30^\circ\)[/tex]:
- The hypotenuse is the longest side, labeled as [tex]\(2\)[/tex].
- The side opposite to [tex]\(30^\circ\)[/tex] is the shortest side, labeled as [tex]\(1\)[/tex].
- The side adjacent to [tex]\(30^\circ\)[/tex] (the side forming the angle with the hypotenuse) is [tex]\(\sqrt{3}\)[/tex].
4. Apply the cosine function:
- The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse.
- Therefore, [tex]\(\cos 30^\circ = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}\)[/tex].
Based on the calculation, [tex]\(\cos 30^\circ\)[/tex] is [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
So, the correct answer is:
E. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
Thus, [tex]\(\cos 30^\circ = 0.8660254037844387\)[/tex], which corresponds to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
