How do you graph polar coordinates?

Question

Quinn Anderson · Accepted Answer

To establish polar coordinates on a plane, we choose a point #O# - the origin of coordinates, the pole, and a ray from this point to some direction #OX# -  the polar axis (usually drawn horizontally).  Then the position of every point #A# on a plane can be defined by two polar coordinates: a polar angle #varphi# from the polar axis counterclockwise to a ray connecting the origin of coordinates with our point #A# - angle #/_ XOA# (usually measured in radians) and by the length #rho# of a segment #OA#. To graph a function in polar coordinates we have to have its definition in polar coordinates. 
Consider, for example a function defined by the formula
#rho=varphi#  for all #varphi>=0#. The function defined by this equality has a graph that starts at the origin of coordinates #O# because, if #varphi=0#, #rho=0#. 
Then, as a polar angle #varphi# increases, the distance from an origin #rho# increases as well. This gradual increase in both polar angle and distance from the origin produces a graph of a spiral.  After the first full circle the point on a graph will hit the polar axis at a distance #2pi#. Then, after the second full circle, it will intersect the polar axis at a distance #4pi#, etc.

Violet Anderson · Answer

undefined To graph polar coordinates, we use the polar coordinate system, which is based on a circle. In this system, each point is determined by a distance from the origin (called the radius, denoted by $r$) and an angle from the positive x-axis (called the angle, denoted by $\theta$).

1. **Plotting Points:** To plot a point given in polar coordinates $(r, \theta)$, start at the origin, move along the polar axis (positive x-axis) to a distance of $r$ units, and then rotate counterclockwise by an angle of $\theta$ degrees.

2. **Special Cases:** 
   - If $r$ is negative, it means the point is on the opposite side of the origin but still at the same angle $\theta$. 
   - If $\theta$ is negative, it means rotating clockwise instead of counterclockwise.

3. **Graphing Curves:** To graph curves given in polar coordinates, such as $r = f(\theta)$, where $f(\theta)$ is some function of $\theta$, you can plot points by choosing different values of $\theta$ and calculating the corresponding $r$ values. Connecting these points will give you the graph of the curve.

4. **Examples:**
   - To graph a circle with radius $r$, you would plot points $(r, \theta)$ for all values of $\theta$ from 0 to $2\pi$.
   - To graph a line through the origin at an angle $\theta$, you would plot points $(r, \theta)$ where $r$ varies.

5. **Conversion:** To convert polar coordinates to Cartesian coordinates, you can use the formulas $x = r \cdot \cos(\theta)$ and $y = r \cdot \sin(\theta)$.

Overall, graphing polar coordinates involves understanding the relationship between the radius and angle and using this information to plot points and graph curves.