Answer :
To solve for \(\cos 60^\circ\), we can utilize our knowledge of trigonometry and special angles.
### Step-by-Step Solution:
1. Understanding Special Angles:
The angle \(60^\circ\) is one of the commonly known special angles in trigonometry. The values of sine, cosine, and tangent for these angles are well-documented.
2. Using the Unit Circle:
In the unit circle, angles are measured from the positive x-axis, and for a \(60^\circ\) angle, we can visualize or recall the specific coordinates of the point on the unit circle that corresponds to this angle.
- The coordinates for \(60^\circ\) are \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\).
3. Definition of Cosine:
The cosine of an angle \( \theta \) in the unit circle is defined as the x-coordinate of the corresponding point.
4. Finding \(\cos 60^\circ\):
Looking at the coordinates \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), the x-coordinate is \(\frac{1}{2}\).
Thus, \(\cos 60^\circ = \frac{1}{2}\).
### Verification Options:
We need to check which option matches our determined value:
A. \(\frac{1}{\sqrt{2}}\)
B. \(\frac{\sqrt{3}}{2}\)
C. \(\frac{1}{2}\)
D. \(\frac{1}{\sqrt{3}}\)
E. \(\sqrt{3}\)
F. 1
Since \(\cos 60^\circ = \frac{1}{2}\), the correct option is:
C. [tex]\(\frac{1}{2}\)[/tex].
### Step-by-Step Solution:
1. Understanding Special Angles:
The angle \(60^\circ\) is one of the commonly known special angles in trigonometry. The values of sine, cosine, and tangent for these angles are well-documented.
2. Using the Unit Circle:
In the unit circle, angles are measured from the positive x-axis, and for a \(60^\circ\) angle, we can visualize or recall the specific coordinates of the point on the unit circle that corresponds to this angle.
- The coordinates for \(60^\circ\) are \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\).
3. Definition of Cosine:
The cosine of an angle \( \theta \) in the unit circle is defined as the x-coordinate of the corresponding point.
4. Finding \(\cos 60^\circ\):
Looking at the coordinates \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), the x-coordinate is \(\frac{1}{2}\).
Thus, \(\cos 60^\circ = \frac{1}{2}\).
### Verification Options:
We need to check which option matches our determined value:
A. \(\frac{1}{\sqrt{2}}\)
B. \(\frac{\sqrt{3}}{2}\)
C. \(\frac{1}{2}\)
D. \(\frac{1}{\sqrt{3}}\)
E. \(\sqrt{3}\)
F. 1
Since \(\cos 60^\circ = \frac{1}{2}\), the correct option is:
C. [tex]\(\frac{1}{2}\)[/tex].
