To determine the value of [tex]\(\cos 45^{\circ}\)[/tex], let's go through the process step-by-step:
1. Understanding the Angle:
- The angle given is [tex]\(45^\circ\)[/tex]. This is a commonly known angle in trigonometry, particularly because it appears in an isosceles right triangle, where both legs are of equal length.
2. Reference Triangle:
- Consider a right triangle where both the angles other than the right angle are [tex]\(45^\circ\)[/tex]. In such a triangle, the lengths of the legs are equal, and the length of the hypotenuse can be derived using the Pythagorean theorem.
3. Isosceles Right Triangle:
- Let's assume the legs of the triangle are each of length 1. Then, by the Pythagorean theorem, the hypotenuse [tex]\(h\)[/tex] would be:
[tex]\[
h = \sqrt{1^2 + 1^2} = \sqrt{2}
\][/tex]
4. Cosine of 45 Degrees:
- The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. For [tex]\(\cos 45^\circ\)[/tex]:
[tex]\[
\cos 45^\circ = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}
\][/tex]
5. Simplifying the Expression:
- The ratio [tex]\(\frac{1}{\sqrt{2}}\)[/tex] is the exact value we use to represent [tex]\(\cos 45^\circ\)[/tex].
6. Decimal Representation:
- The value [tex]\(\frac{1}{\sqrt{2}}\)[/tex] can be approximated as a decimal for more practical purposes, which is approximately [tex]\(0.7071067811865475\)[/tex].
Given these steps and understanding the fundamental trigonometric properties, we can conclude that the value of [tex]\(\cos 45^\circ\)[/tex] is:
D. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
