The graph can be considered the set of all solutions to the Pythagorean theorem where is set at Several points on the graph (with approximations) are shown above. Also, note that
What does the entire graph of look like?
In a traditional -coordinate system from the previous problem, the first value indicates horizontal distance and the second value indicates vertical distance. In a polar coordinate system using the value gives "distance from the origin when facing angle " Some example points are shown above.
One curious thing is that can be negative. Which point is
We saw the graph of the circle above written as in rectangular coordinates. How can you write it using polar coordinates?
Other than the same point being identified with a positive (facing forward) or a negative (facing backward), it's possible for the same point to be identified in an infinite number of ways via the measure of angle.
The point below can be identified as Which of the choices also identifies the same point?
In the visualization above, is restricted to only negative values. It can still be used to help to answer the question "Which pair of coordinates refer to the same point?"
Because of possible negative values, polar graphs are often restricted to particular values of and
If we restrict to positive values only, what kind of graph will be?
The visualization below lets you see the effect of in the spiral graph
What will happen to the graph if is allowed to be negative?
Polar coordinates give the flexibility to include whole new classes of graphs, like the flower with overlapping petals above. Many other marvels await: start now with an intuitive look at the six main functions of trigonometry.