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Why sine ratio is the ratio of opposite side and hypotenuse?
Because that is accepted definition. The sine is opposite over hypotenuse, or Y in the unit circle. The cosine is adjacent over hypotenuse, or X in the unit circle. The tangent is sine over cosine, etc. For more information, please see the related link below.
What is a sine in mathematics?
A function that depends on the value of an angle. One way to define it is with a unit circle (a circle with center in the coordinate origin, and radius of 1). To the right is zero, from there, a positive angle is counterclockwise. In this case, the sine is simply the y-coordinate, and the cosine is the x-coordinate of the point on the circle where the ray of the angle crosses the circle. The value of the sine (and cosine) obviously depends on the angle - that's why it is considered a "function". Sine, cosine, tangent, cotangent, cosecans, and secans can also be defined via right triangles; for more details see here: http://en.wikipedia.org/wiki/Sine#Sine.2C_cosine_and_tangent
How the sine curve related to a curve?
Basically, it IS a curve.
On the unit circle define the sine function by using the distance walked as the input and the y coordinate as the output?
Yes, but only if the argument of the sine function is in radians.
Why are sine functions such good models for repetitive behavior?
Repetitive behavior can be described by a point moving in a circle. The time of repetition is equivalent to time taken by that particle to complete that circle. When the point moves in a circle, its angle changes from 0 to 360 degrees; all of these values can be given by a sine function or a cosine function.
