## Trigonometry, polar and cartesian co-ordinates, and circular motion.

We can give the position of a point in two dimensions in one of two ways: we can use *cartesian* co-ordinates, x and y, or we can use *polar* co-ordinates, ρ (rho) and θ (theta).

Cartesian co-ordinates are named after the French mathematician René Descartes (1596 - 1650). The position of a point is given by its distance from the two axes (almost invariably called X and Y).

In polar co-ordinates the position of a point is given by its distance from the origin (ρ or rho) and the angle the line from the point to the origin makes with a horizontal line through the origin (θ or theta). The length of the line is always positive and (in mathematics) the angles are measured anti-clockwise.

*Compass bearings are usually measured clockwise from North but are not considered further here.*

The relationship between the cartesian and polar co-ordinates of a point is given by

x = rho cos theta

y = rho sin theta

We can see that *sin theta = cos (90° - theta)*

In the First Quadrant theta is in the range 0° to 90°. In the Second Quadrant theta is in the range 90° to 180°.

In the Second Quadrant y is still positive but x is negative so *sin theta* is positive but *cos theta* is negative.

We can see that *sin theta = sin (180° - theta) and cos theta = - cos (180° - theta)*

In the Third Quadrant theta is in the range 180° to 270°.

In the Third Quadrant both x and y, and so both *sin theta* and *cos theta*, are negative.

In the Fourth Quadrant theta is in the range 270° to 360°. Here y is negative but x is positive so *sin theta* is negative and *cos theta* is positive.

Here are curves of sin theta and cos theta for theta from -90° to +450°. The pattern is repeated every 360°, so sin (theta + 360°) = sin theta, cos (theta + 360°) = cos theta etc .

The graph covers values of theta from -90° to +450° to make it easier to see that although the sine and cosine curves have the same shape, called a *sine wave*, they are 90° apart (are 90° out of phase).

Although for each value of theta there is only one value of *sin theta* and one of* cos theta*, for each value of *sin theta* and *cos theta* there are two values of theta (more of course if you include angles less than 0° or greater than 360°), and this is important when we are using trigonometry to calculate the sides and angles of a triangle.

© Barry Gray March 2008