Graphs of Cosine and Sine
Author: Jamie Mulholland

Cosine and sine of an angle θ are defined to be the x and y-coordinates of the point P on the unit circle with the property that the line from O to P makes and angle θ with the positve x-axis.

The applet below shows how the graphs of sin(θ) and cos(θ) follow directly from the definition of these functions.  Move the slider to change the angle.  Observe the point moves once around the circle it returns to its original position - this is the reason cos and sin are periodic.

sine

cosine

y=sin(\theta)

y=cos(\theta)

T = 1 (slider from 0 to 10, step 0.1)
CircleCenter = (-2, 0)
P: CircleCenter + (cos(T), sin(T))
UnitCircle = Circle(CircleCenter, 1)
B = Intersect(UnitCircle, xAxis, 2)
CurveSin = Function(sin(x), 0, T)
CurveCos = Function(cos(x), 0, T)
A = Point(Circle[CircleCenter, x(P)])
PoinOnAxis: (x(P), 0)
SegmentX = Segment(CircleCenter, PoinOnAxis)
SegmentY = Segment(PoinOnAxis, P)
TipSin: (T, sin(T))
TipCos: (T, cos(T))
SegmentCircleToSin = Segment(P, TipSin)
BoolSine = true
BoolCosine = false
RadialSegment = Segment(CircleCenter, P)
θ = Angle(B, CircleCenter, P)
T3: "θ = " + T + " rad"