Polar Coordinates

When dealing with certain functions, the system that we are accustomed to becomes
inconvenient and troublesome. We usually use rectangular coordinates, that is, coordinates
using the x and the y axis to plot points and describe functions, but certain functions
get very complicated if we try to use these coordinates on the Cartesian graph.

Before looking at polar coordinates, it is important to understand the functions of
sine, cosine, and tangent and how they are used to find a given
angle in a right triangle, because we will be doing these calculations quite often.

In polar coordinates, we describe points using their distance (r) from
the origin and the angle (θ) from the positive horizontal axis. Similar
to the x and y coordinate, the distance and angle are called the radial
and angular coordinate.

Polar coordinates are then written as (r,θ). The origin is now called
the pole, and the x axis is called the polar axis, because every angle is dependent on it.

The angle measurement θ can be expressed in radians or degrees. If we
recall from circles in Geometry, a radian is the measure of the radius around the circumference
of the circle, and 2Π radians is how many radians it takes to go around the circle. Similarly,
360 is how many degrees it takes to make a full circle.

(1) The point described by the polar coordinate (3, Π/4) will look like

Rectangular (or Cartesian) coordinates are defined in terms of polar coordinates by

Conversely, we can define polar coordinates in terms of x and y

One of these should look familiar. It makes sense that the radius is defined in terms
of x and y this way because it is the same as the pythagorean theorem. Every r can be expressed
as the hypotnuse of a right triangle formed by rectangular coordinates.

(2) The rectangular coordinate (3,5) will be translated

The quadrant of the angle as well as the signs of x and y can determine if the radius
is positive or negative.

We must remember that the sign of the radius flips the quadrant of the angle. For example,
the point (2,Π/6) is the same as the point (-2,7Π/6)

Converting Degrees and Radians

The angle value of a polar coordinate can be given in either degress or radians.
Converting either of them is a one step process involving proportions.

Since 360 degrees is the same as a radian length of 2pi, we can set them equal and
have one given theta as well as one that we are solving for.

Here is a table of common angle measurements in radians and degrees. After a while,
it can be beneficial to memorize which common degree and radian measurements correspond
to each other.

We must remember that there are an infinite amount of ways to describe an angle, because
we can always add or subtract 2Π from an angle to get the same measurement. For instance

Polar Equations

(3) Let’s graph the equation

We can see from this equation that our radius depends on our input, which is the angle.
We can make a chart similar to an xy chart and see what outputs we will get.

We can see that the image forms a smooth circle.

(4) Let’s try converting the following equation in rectangular form to polar form.

We can rewrite the equation as

This allows us to use the equalities giving the relationship between polar and
rectangular coordinates and substitute them in for x and y.

(5) We can also convert equations in polar form to rectangular form

We can multiply both sides by r and use the equalities to substitute

We can complete the square and end up with the equation for a circle

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