To find the exact value of cos 45°, we can use the known properties of a 45°-45°-90° triangle. In a 45°-45°-90° triangle, the two legs are of equal length, and the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of one of the legs.
Let's say each leg of the triangle has a length of 1. Then, by the Pythagorean theorem, the hypotenuse will be:
[tex]\[ \text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2} \][/tex]
The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. For a 45° angle in this triangle, the adjacent side is one leg of the triangle, and the hypotenuse is [tex]\( \sqrt{2} \)[/tex].
Thus, the cosine of 45° is:
[tex]\[ \cos 45° = \frac{1}{\sqrt{2}} \][/tex]
To express this in a simpler form, we can rationalize the denominator:
[tex]\[ \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
Therefore, the exact value of cos 45° is:
[tex]\[ \boxed{\frac{\sqrt{2}}{2}} \][/tex]
