To determine the value of [tex]\(\cos 45^\circ\)[/tex], we need to consider the properties of the angle in question.
The angle [tex]\(45^\circ\)[/tex] is a special angle in trigonometry, often encountered in a right-angled isosceles triangle. In such a triangle, the two non-hypotenuse sides are of equal length, and the angles opposite these sides are both [tex]\(45^\circ\)[/tex].
The function [tex]\(\cos(\theta)\)[/tex] for an angle [tex]\(\theta\)[/tex] in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse.
For [tex]\(45^\circ\)[/tex]:
- In a right-angled isosceles triangle, if each of the legs (adjacent and opposite sides) is of length 1, the hypotenuse can be found using the Pythagorean theorem:
[tex]\[
\text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2}
\][/tex]
- The cosine of [tex]\(45^\circ\)[/tex] is then:
[tex]\[
\cos 45^\circ = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}
\][/tex]
To rationalize the denominator, we multiply the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
Thus, the correct answer for [tex]\(\cos 45^\circ\)[/tex] in terms of its numerical value is:
[tex]\[
\cos 45^\circ \approx 0.7071067811865476
\][/tex]
Given the multiple-choice options:
A. [tex]\(\frac{1}{2}\)[/tex]
B. 1
C. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
D. [tex]\(\sqrt{2}\)[/tex]
The value closest to our calculated result and its simplified form [tex]\(\frac{1}{\sqrt{2}}\)[/tex] is:
[tex]\[
\boxed{\frac{1}{\sqrt{2}}}
\][/tex]
