To find [tex]\(\cos 45^{\circ}\)[/tex]:
1. Use the Unit Circle or Trigonometric Table:
- The angle [tex]\(45^{\circ}\)[/tex] is a well-known angle in trigonometry.
- On the unit circle, [tex]\(\cos 45^{\circ}\)[/tex] corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle.
2. Basic Trigonometric Identity for Special Angles:
- For [tex]\(45^{\circ}\)[/tex], the coordinates of the point are [tex]\(\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)[/tex].
3. Cosine of [tex]\(45^{\circ}\)[/tex]:
- The cosine function gives the x-coordinate of the point on the unit circle.
- Therefore, [tex]\(\cos 45^{\circ} = \frac{1}{\sqrt{2}}\)[/tex].
4. Simplifying the Result:
- Note that [tex]\(\frac{1}{\sqrt{2}}\)[/tex] can be rationalized to [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
Thus, [tex]\(\cos 45^{\circ} = \frac{1}{\sqrt{2}}\)[/tex], which is approximately [tex]\(0.7071067811865476\)[/tex].
Given the options:
A. 1
B. [tex]\(\sqrt{2}\)[/tex]
C. [tex]\(\frac{1}{2}\)[/tex]
D. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
The correct answer is:
[tex]\[ \boxed{\frac{1}{\sqrt{2}}} \][/tex]
