naively, sine of an angle, is the ratio of the length of the perpendicular to the length of the hypotenuse of the right angled triangle that this angle creates.
So, in a 45-45-90 triangle, the sides are 'a', 'a' and 'sqrt(2)*a.
So according to the above definition, sin(45) = a/sqrt(2)*a = 1/sqrt(2)
I just dont see why the ratio of the lengths of a triangle would be beneficial.... so once I find the sin, cos, tan, or their recipricols of an angle.... how is that ratio going to help me?
Well, firstly we see that trigonometric ratios, even though defined with respect to a particular triangle, it is independent of the size of the triangle (in my example, 'a').
In fact, as you learn more, the dependency on a right angle triangle in defining sine and cos is completely removed. Given this, sines and cos retain the properties of the triangle without having anything to do it.
Trigonometric Functions have extreme importance in topics of vector analysis and calculus because of their very special properties. However, its major application comes in fourier analysis, because of a theorem due to fourier which states that
'every periodic function, can be written as a linear combination of sines and cosines.'
So you see, even though we begin with defining it as a ratio, it develops into something much more, which helps us develop several other areas of mathematics.