What is the value of sin 45°?

The value of sin 45° is approximately 0.707. This can be understood by considering the properties of a 45°-45°-90° triangle, where the two non-hypotenuse sides are equal. If we denote the length of each of these sides as 1, the length of the hypotenuse can be calculated using the Pythagorean theorem:

[ \text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2} ]

The sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, for a 45° angle, the opposite side has a length of 1 and the hypotenuse, as calculated, has a length of (\sqrt{2}). Therefore, we have:

[ \sin 45° = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}} ]

To express (\frac{1}{\sqrt{2}}) in decimal form, it is approximately equal to 0.707. Thus, the value 0