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# Rectangular to Polar Conversions

## Convert from cartesian to polar coordinates

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Practice Rectangular to Polar Conversions
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Polar Graph Conversions

### Vocabulary

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:

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

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

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

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

Write each polar equation in rectangular form:

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

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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 π\begin{align*}\pi\end{align*} radians must be added to the value of θ\begin{align*}\theta\end{align*} for all points with rectangular coordinates in the second and third quadrants.

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Write the following points, given in rectangular form, in polar form using radians where0θ2π\begin{align*}0\leq \theta \leq 2\pi\end{align*} .

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 x=rcosθ\begin{align*}x = r \cos \theta\end{align*} and y=rsinθ\begin{align*}y = r \sin \theta\end{align*}.

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