What is the value of cos 45°?

The value of cos 45° is derived from the properties of a 45-45-90 triangle, which is an isosceles right triangle. In this type of triangle, both angles are 45 degrees, and the lengths of the legs are equal. By applying the Pythagorean theorem, one can determine that the length of the hypotenuse is equal to the square root of 2 times the length of each leg.

When you calculate the cosine of an angle in a right triangle, it is defined as the ratio of the length of the adjacent side to the hypotenuse. For a 45-45-90 triangle, if each leg is of length 1, the hypotenuse will be ( \sqrt{2} ). Therefore, the cosine of 45° becomes:

[ \cos 45° = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0.707. ]

Thus, the correct answer, which approximates the ratio, is 0.707. This value is commonly used in trigonometry and is essential