Answer :
To determine the value of [tex]\(\cos 45^{\circ}\)[/tex], we need to recall a basic fact from trigonometry:
1. The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the hypotenuse.
2. For a 45-degree angle in such a triangle, the two non-hypotenuse sides are equal because it is an isosceles right triangle (also known as a 45-45-90 triangle).
Given the properties of the 45-45-90 triangle:
- The lengths of the legs are equal.
- The length of the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
To be more specific, if the legs are each of length [tex]\(1\)[/tex]:
- The hypotenuse [tex]\( h \)[/tex] will have a length of [tex]\( h = \sqrt{1^2 + 1^2} = \sqrt{2} \)[/tex].
Therefore, we find:
[tex]\[ \cos 45^{\circ} = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}. \][/tex]
Note: The value [tex]\(\frac{1}{\sqrt{2}}\)[/tex] can also be written in its rationalized form, which would be:
[tex]\[ \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
However, looking at the answer choices provided:
- D. [tex]\( \frac{1}{\sqrt{2}} \)[/tex]
Thus, the value of [tex]\(\cos 45^{\circ}\)[/tex] is:
D. [tex]\(\frac{1}{\sqrt{2}}\)[/tex].
After verifying the provided answer, it matches our obtained value:
[tex]\[ \cos 45^{\circ} = 0.7071067811865476. \][/tex]
So, the correct choice from the options is indeed:
D. [tex]\(\frac{1}{\sqrt{2}}\)[/tex].
1. The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the hypotenuse.
2. For a 45-degree angle in such a triangle, the two non-hypotenuse sides are equal because it is an isosceles right triangle (also known as a 45-45-90 triangle).
Given the properties of the 45-45-90 triangle:
- The lengths of the legs are equal.
- The length of the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
To be more specific, if the legs are each of length [tex]\(1\)[/tex]:
- The hypotenuse [tex]\( h \)[/tex] will have a length of [tex]\( h = \sqrt{1^2 + 1^2} = \sqrt{2} \)[/tex].
Therefore, we find:
[tex]\[ \cos 45^{\circ} = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}. \][/tex]
Note: The value [tex]\(\frac{1}{\sqrt{2}}\)[/tex] can also be written in its rationalized form, which would be:
[tex]\[ \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
However, looking at the answer choices provided:
- D. [tex]\( \frac{1}{\sqrt{2}} \)[/tex]
Thus, the value of [tex]\(\cos 45^{\circ}\)[/tex] is:
D. [tex]\(\frac{1}{\sqrt{2}}\)[/tex].
After verifying the provided answer, it matches our obtained value:
[tex]\[ \cos 45^{\circ} = 0.7071067811865476. \][/tex]
So, the correct choice from the options is indeed:
D. [tex]\(\frac{1}{\sqrt{2}}\)[/tex].
