NUMBAT OER - Open Educational Resources

3. Going round in circles

The radian unit links the angle and its trigonometric functions directly to a circle. If you have a point moving at a constant distance from a second, fixed point, the path of the first point describes a circle. We can describe the location of the point in terms of the angle that the line joining the two points (the radius) makes with the vertical. For a distance from the centre of r, the x- and ycoordinates of the moving point are given by:

x = r sinα
y = r cosα

where the angle α is measured clockwise from the vertical (ie the y-axis). So if we plot the cosine of an angle against its sine, we draw a perfect circle with a radius of one unit (r = 1):

A circle defined by the relationship between cosine and sine

You can see that for angles up to 0.5π radians (90°), both the sine and cosine have positive values. Between 0.5π radians (90°) and 1π radians (180°), the sine is still positive but the cosine is negative. From 1π radians (180°) to 1.5π radians (270°), both are negative, and then from 1.5π radians (270°) to 2π radians (360°) the sine remains negative but the cosine is positive.