What is the definition of sine (sin) in a right triangle?

In a right triangle, the sine (sin) of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. This relationship is fundamental in trigonometry and is particularly useful for solving problems involving right triangles.

When you have a right triangle with one angle designated as the angle of interest, the hypotenuse is the longest side of the triangle, which is always opposite the right angle. The opposite side is the one that is directly across from the angle you are considering. Hence, when taking the sine of that angle, you are calculating how the length of the opposite side compares to the hypotenuse. This ratio helps to relate the angles and sides of the triangle and is vital for various applications in physics and engineering, such as in wave mechanics and oscillations, where angles frequently arise.