To find the value of [tex]\(\cos 45^\circ\)[/tex], we can follow the principles of trigonometry and the properties of angles in a right-angled triangle.
First, let's understand that [tex]\(45^\circ\)[/tex] is a special angle in trigonometry. In the context of a right-angled triangle, an angle of [tex]\(45^\circ\)[/tex] signifies an isosceles right-angled triangle, where the two non-right angles are equal, both being [tex]\(45^\circ\)[/tex].
In such a triangle, let's denote the lengths of the legs adjacent and opposite to the [tex]\(45^\circ\)[/tex] angle as [tex]\(a\)[/tex]. According to the Pythagorean theorem:
[tex]\[
a^2 + a^2 = ( \text{hypotenuse} )^2
\][/tex]
Simplifying this, we get:
[tex]\[
2a^2 = (\text{hypotenuse})^2
\][/tex]
[tex]\[
\text{hypotenuse} = a \sqrt{2}
\][/tex]
The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent leg to the length of the hypotenuse. Therefore, for [tex]\( \cos 45^\circ \)[/tex]:
[tex]\[
\cos 45^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{a\sqrt{2}} = \frac{1}{\sqrt{2}}
\][/tex]
To double-check our understanding, we can convert the fraction [tex]\(\frac{1}{\sqrt{2}}\)[/tex] into its decimal form. The decimal value for this ratio is approximately:
[tex]\[
\cos 45^\circ \approx 0.707
\][/tex]
This matches the known value of [tex]\(\cos 45^\circ\)[/tex].
Given the options, the correct answer is:
[tex]\[
\cos 45^\circ = \frac{1}{\sqrt{2}}
\][/tex]
So, the correct option is:
[tex]\[
D. \frac{1}{\sqrt{2}}
\][/tex]
