M408M Learning Module Pages
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Chapter 10: Parametric Equations
and Polar Coordinates
Learning module LM 10.1:
Parametrized Curves:
Learning module LM 10.2: Calculus with Parametrized Curves:
Learning module LM 10.3: Polar Coordinates:
Definitions of polar coordinates
Graphing polar functions
Computing slopes of tangent lines
Learning module LM 10.4: Areas and Lengths of Polar Curves:
Learning module LM 10.5: Conic Sections:
Learning module LM 10.6: Conic Sections in Polar Coordinates:
Chapter 12: Vectors and the Geometry of Space
Chapter 13: Vector Functions
Chapter 14: Partial Derivatives
Chapter 15: Multiple Integrals


Definitions of polar coordinates
Definitions of Polar Coordinates
The polar coordinates of the point $P$ shown
to the right are written $(r,\, \theta)$ where $r$ is the distance of
$P$ from the origin and $\theta$ is the angle the line from the origin
to $P$ makes with the $x$axis, the angle being measured as $\theta$
rotates counterclockwise starting from the positive $x$axis. At the
origin $r= 0$, but $\theta$ is not welldefined, so we usually assign
to the origin the polar coordinates $(0,\, \theta)$ for any choice of
$\theta$.


By righttriangle trigonometry and the
Pythagorean theorem: $$ \sin \theta \ = \
\frac{y}{r}, \quad \cos \theta \ = \ \frac{x}{r}\,, \quad r^2 \ = \
x^2 + y^2\,, \quad \tan \theta \ = \ \frac{y}{x}\,.$$ Thus for
a point $P$ the Cartesian and polar coordinate systems are related
by $$ (x,\, y) \ = \ (r \cos \theta,\, r \sin
\theta)\,, \quad (r,\, \theta) \ = \ \left(\sqrt{x^2+y^2},\
\tan^{1}\Bigl( \frac{y}{x}\Bigr)\right)\,.$$ 

Notice that the choices of coordinates $r$ and $\theta$ are not
unique for a given point $P$ since $(r,\, \theta),\ (r,\, \theta
+2\pi)$ and $(r,\, \pi +\theta)$ all describe the same point $P$.
