Sine and Cosine
Sine and Cosine: Definition
The sine and cosine functions can be defined in a number of ways:
Definition I: From a triangle
Given any angle q (0 £ q £ 90°), we can find the sine or cosine of that angle by constructing a right triangle with one vertex of angle q. The sine is equal to the length of the side opposite to q, divided by the length of the triangle's hypotenuse. The cosine is equal to the length of the side adjacent to q, divided by the length of the triangle's hypotenuse. In this way, we can find the sine or cosine of any q in the range 0 £ q £ 90°.
Definition II: From the unit circle
Draw a unit circle, in that a circle of radius 1, centered at the origin of a 2-dimensional coordinate system. Given an angle q, locate the point on the circle that is located at an angle q from the origin. (According to standard convention, angles are measured counter-clockwise from the positive horizontal axis.) The sin(q) can be defined as the y-coordinate of this point. The cos(q) can be defined as the x-coordinate of this point. In this way, we can find the sine or cosine of any real value of q (q Î Â).
Definition III: An algebraic approach
From defining a few general properties of the sine and cosine functions, we can alegbraically derive the sine and cosine functions themselves.
Definition IV: Over the complex numbers
Given a complex number z = a + b i,
sin(z) = sin(a)cosh(b) + cos(a)sinh(b) i