The General Angle
Consider a radius of length '1' rotating anti-clockwise about the origin. The coordinates of any point on the circle give the values of the adjacent and opposite sides of a right angled triangle, with the radius the hypotenuse.
The General Angle (theta) is the included angle between the radius and the x-coordinate of the point.
As the radius rotates the x and y values change. Hence the values of sine, cosine and tangent also change.
The result is summarized in the diagram below.
Example #1
Example #2
Sine
- the sine graph starts at zero
- it repeats itself every 360 degrees(or 2 pi)
- y is never more than 1 or less than -1 (displacement from the x-axis is called the amplitude)
- a sin graph 'leads' a cos graph by 90 degrees
Cosine
- the cosine graph starts at one
- it repeats itself every 360 degrees(or 2 pi)
- y is never more than 1 or less than -1 (displacement from the x-axis is called the amplitude)
- a cos graph 'lags' a sin graph by 90 degrees(pi/2) - this is termed a phase shift
Tangent
- the tangent graph starts at zero
- it repeats itself every 180 degrees
- y can vary between numbers approaching infinity and minus infinity
Further comparison
- only the cosine function is symmetrical about the y-axis
- all the functions are cyclic - the distance along the horizontal axis repeated is called the period
