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Polar to Rectangular Conversions

Convert from polar to cartesian coordinates

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Polar Graph Conversions

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Explain how to graph rectangular coordinates: _______________________________________________________________________

Explain how to graph  polar coordinates:


Polar to Rectangular Conversion

We can use trigonometry to convert from polar coordinates to rectangular coordinates.

Complete the coordinate conversion equations:

\begin{align*}x =\end{align*}x= _________________                \begin{align*}y =\end{align*}y=_________________

\begin{align*}r^2 =\end{align*}r2=_________________                \begin{align*}tan\theta =\end{align*}tanθ=_________________


Given the following polar coordinates, find the corresponding rectangular coordinates of the points:

  1. \begin{align*}(3, \frac{\pi}{3})\end{align*}(3,π3)
  2. \begin{align*}(2, \frac{3\pi}{2})\end{align*}(2,3π2)
  3. \begin{align*}(5, \frac{\pi}{4})\end{align*}(5,π4)

Write each polar equation in rectangular form:

  1. \begin{align*}r=4\cos \theta \end{align*}r=4cosθ
  2. \begin{align*}r=10\sin \theta \end{align*}r=10sinθ
  3. \begin{align*}r=5\csc \theta \end{align*}r=5cscθ


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Rectangular to Polar Conversion

To convert from rectangular to polar coodinates, we use the Pythagorean Theorem and the Arctangent function.

Note: The Arctangent function only calculates angles in the first and fourth quadrants so \begin{align*}\pi\end{align*}π radians must be added to the value of \begin{align*}\theta\end{align*}θ for all points with rectangular coordinates in the second and third quadrants.


Write the following points, given in rectangular form, in polar form using radians where\begin{align*}0\leq \theta \leq 2\pi\end{align*}0θ2π .

Remember: There are many possible polar coordinates!

  1. (5,3)
  2. (-2,4)
  3. (-7,1)
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For each equation, convert the rectangular equation to polar form.

Hint: Use the equations \begin{align*}x = r \cos \theta\end{align*}x=rcosθ and \begin{align*}y = r \sin \theta\end{align*}y=rsinθ.

  1. \begin{align*}2x-y=6\end{align*}2xy=6
  2. \begin{align*}3x+4y=2\end{align*}3x+4y=2
  3. \begin{align*}(x+2)^2+y^2=4\end{align*}(x+2)2+y2=4
  4. \begin{align*}(x+5)^2+(y-1)^2=26\end{align*}(x+5)2+(y1)2=26
  5. \begin{align*}x^2+(y-6)^2=36\end{align*}x2+(y6)2=36
  6. \begin{align*}x^2+(y+2)^2=4\end{align*}x2+(y+2)2=4
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