Sine, Sinusoids and Complex Numbers
Sine and Cosine
Sine and cosine are trigonometric functions of an angle. The sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent side to that of the hypotenuse.
The sine can be interpreted as the height above the diameter on the circumference at angle θ in a unit circle. In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0)
The trigonometric functions cosine and sine of angle θ may be defined on the unit circle as follows: if ( x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to ( x, y) makes an angle θ from the positive x-axis, (where counterclockwise turning is positive), then
- cos θ = x and sin θ = y
Using the Pythagorean theorem, we find that x² + y² = 1, which establishes the fundamental relation cos²θ + sin²θ = 1.
Moreover, observing the periodicity of cosine and sine functions, we note that they repeat their values after each full rotation (360 degrees or 2π radians) around the unit circle. This periodicity can be expressed by the identities cos θ = cos(2πk + θ) and sin θ = sin(2πk…
