Helpful Tools for Calculus, Chapter 1: Functions and Relations
Previous Section: 1.14 Parametric Equations
The Polar coordinate system is an alternative system which measures points in terms of a radius and an angle, f(r,θ), as opposed to with rectangular coordinates, f(x,y). All measurements are done in reference to the origin. The angle is measured in the same way that the unit circle is, so 0 radians is to the right, while π radians is to the left. The angle describes the direction the point is located in. The radius describes the distance to that point.
Plotting and Converting Points
Example 1: Plot the polar coordinate pairs (2,π/3) and (-1,3π/4).
Solution:
Example 2: Convert from rectangular to polar coordinates (3,√3)
Solution: By drawing plotting this on a coordinate plane and connecting the sides like a triangle we get the following diagram:
r can be calculated by finding the hypotenuse of the triangle and the angle of elevation from the origin can be described using tangent.
As we can see from the previous example, we can describe the radius of the polar coordinate system using the Pythagorean Theorem: 
The angle can then be calculated be writing it as a ratio of the x and y coordinates using tangent: tanθ=y/x, as long as x≠0. Use inverse trigonometry to isolate the angle as θ=arctan y/x.
Example 3: Convert from rectangular to polar coordinates (-4,2)
Solution:
There is an easy way to describe a polar coordinate in terms of its rectangular counterpart. If we use our knowledge of trigonometry we can construct the following triangle for any point:
It can thus be shown the x-coordinate could be described using trigonometry as x=rcosθ and, similarly, y=rsinθ.
Example 4: Convert from polar coordinate to rectangular coordinates (3,π/6)
Solution:
Example 5: Convert from polar coordinate to rectangular coordinates (-4,π/3)
Solution:
Graphing and Polar Regions
As preparation for future chapters, it would be helpful for us to tackle identifying regions and graphing lines in polar coordinates. In order to identify a region of space, we can describe it in terms of inequalities for where the region begins and ends, both in terms of angle/direction as well as the distance from the origin.
Example 6: Describe the shaded sector with inequalities describing r and θ:
Solution: Beginning with the radius, we can see that the shaded region lies outside of the circle with radius 3 and inside the circle of radius 5. So, 3≤r≤5. In terms of angles, the minimum angle is π/6 while the larger angle is 
Example 7: Write the polar equation for a line that goes through the origin with a slope of 3/5.
Solution: The slope being 3/5 is the same as having a triangle whose rise is 3 and run is 5. So, the slope, given by the ratio of the sides of a triangle, can be found using tanθ=m. Solving for θ:
θ=arctan(3/5)
θ=0.1720π
Since the line that goes through the origin doesn’t have a restriction on its ‘length’ we know that r has no place in the polar equation. Thus, θ=0.1720π is the entire polar equation.
Example 8: Convert the polar equation r=2secθ into rectangular coordinates.
Solution: Using our substitutions from previous examples and our knowledge of trigonometry,
Example 9: Convert the polar equation r(cosθ+sinθ)=4 into rectangular coordinates.
Solution:
