Calculus
Lesson
61
Polar
Coordinates
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Similar to what we just did for parametric equations:
Examples of simple shapes:
Circle: 
Circle: 
Line: 
Vertical Line: 
Spiral: 
Some hints: if equation does not change when 
, x-axis symmetry
if equation does not change when 
,
y-axis symmetry
if equation does not change when 
,
origin symmetry
To sketch: If there is no easy conversion to rectangular coordinates and
figuring out what it should look like is not easy, try
making a table of 
values.
See shapes above for examples, plus 
and 
.
As an aid to sketching, find slope! We know that 
, so
So we can find 
!
For tangents: 
,
it's a horizontal line and vice-versa.
Examples:
Find all horizontal and vertical tangents:
Other tangent lines: If 
and
then the line 
is
a tangent line through the origin. (we can derive this from our definition of
dy/dx). So, whenever 
,
if there is a non-zero derivative, then we have
a tangent line!
Example: Find all of the tangent lines (both through the origin and
horizontal and vertical) of 