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Rewrite the polar equation $\text{\hspace{0.17em}}r=\frac{3}{1-2\mathrm{cos}\text{\hspace{0.17em}}\theta}\text{\hspace{0.17em}}$ as a Cartesian equation.
The goal is to eliminate $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r,$ and introduce $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ We clear the fraction, and then use substitution. In order to replace $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y,$ we must use the expression $\text{\hspace{0.17em}}{x}^{2}+{y}^{2}={r}^{2}.$
The Cartesian equation is $\text{\hspace{0.17em}}{x}^{2}+{y}^{2}={\left(3+2x\right)}^{2}.\text{\hspace{0.17em}}$ However, to graph it, especially using a graphing calculator or computer program, we want to isolate $\text{\hspace{0.17em}}y.$
When our entire equation has been changed from $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}$ we can stop, unless asked to solve for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ or simplify. See [link] .
The “hour-glass” shape of the graph is called a hyperbola . Hyperbolas have many interesting geometric features and applications, which we will investigate further in Analytic Geometry .
Rewrite the polar equation $\text{\hspace{0.17em}}r=2\mathrm{sin}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in Cartesian form.
${x}^{2}+{y}^{2}=2y\text{\hspace{0.17em}}$ or, in the standard form for a circle, $\text{\hspace{0.17em}}{x}^{2}+{\left(y-1\right)}^{2}=1$
Rewrite the polar equation $\text{\hspace{0.17em}}r=\mathrm{sin}\left(2\theta \right)\text{\hspace{0.17em}}$ in Cartesian form.
This equation can also be written as
Access these online resources for additional instruction and practice with polar coordinates.
Conversion formulas | $\begin{array}{ll}\hfill & \mathrm{cos}\text{\hspace{0.17em}}\theta =\frac{x}{r}\to x=r\mathrm{cos}\text{\hspace{0.17em}}\theta \hfill \\ \hfill & \mathrm{sin}\text{\hspace{0.17em}}\theta =\frac{y}{r}\to y=r\mathrm{sin}\text{\hspace{0.17em}}\theta \hfill \\ \hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{r}^{2}={x}^{2}+{y}^{2}\hfill \\ \hfill & \mathrm{tan}\text{\hspace{0.17em}}\theta =\frac{y}{x}\hfill \end{array}$ |
How are polar coordinates different from rectangular coordinates?
For polar coordinates, the point in the plane depends on the angle from the positive x- axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.
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