The sine of 60 degrees is a well-known value in trigonometry. To find sin(60°), we can relate it to the properties of a 30-60-90 triangle. In such a triangle, the sides have specific ratios: the length of the side opposite the 30° angle is half the length of the hypotenuse, while the side opposite the 60° angle is √3 times the length of the side opposite the 30° angle.
For a 30-60-90 triangle, if we designate the hypotenuse as 2, the lengths of the respective sides would be:
- Opposite the 30° angle:
1 - Opposite the 60° angle:
√3
Thus, to calculate the sine of 60 degrees, we use the definition of sine in a right triangle, which is the ratio of the length of the side opposite the angle to the length of the hypotenuse:
sin(60°) = rac{opposite}{hypotenuse} = rac{ ext{Side opposite 60°}}{ ext{Hypotenuse}} = rac{√3}{2}
Therefore, the exact value of sin(60°) as a simplified fraction is: