What is Sine?

Back in the school days, we were taught that Sine is a Trigonometric function of an angle. In the context of a right-angled triangle, it is the ratio of the length of the side opposite that angle to the length of the hypotenuse. That was pretty much about it! With the help of some Trigonometric formulas, we were able to solve a whole lot of Math problem. But some questions remain unanswered - What exactly is Sine? Where did all of this come from?

In this article, we'll try to answer these questions in an easy to understand manner.. First, let us begin with a unit circle. A unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.Something like this:

Now, let us make a right-angled triangle inside this unit circle.

The point A on the circle has co-ordinates (x,y). How can we know their value? As stated earlier, Sine is defined as ratio of the length of the side opposite that angle to the length of the hypotenuse.Similarly, Cosine is defined as the ratio of the lengths of the side of the triangle adjacent to the angle and the hypotenuse. Going by these definitions, we get x=cosθ,y=sinθ$x=cos\theta ,y=sin\theta$

So, if we have y = sin x and rotate it along the circle, we'll get our sine graph!

Similarly, if we got about evaluating the co-ordinates in the circle at various angles, we will get all the values for Sine and Cosine!

Taking it into Differential Calculus

Now, we have studied that d/dx(sin x ) = cos x . Where did this come from? In differential calculus, the derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point.

In Sine graph, the slope at π2$\frac{\pi }{2}$ and 3π2$\frac{3\pi }{2}$ is equal to zero. So, if we take out the slope at each point, we'll get the cosine function. So that's why, the derivative of sin x is equal to cos x.