Polar Coordinates

Figure 1

Plotting Points Using Polar Coordinates

Figure 2

Solution

The angle $\frac{\pi }{2}$ is found by sweeping in a counterclockwise direction 90° from the polar axis. The point is located at a length of 3 units from the pole in the $\frac{\pi }{2}$ direction, as shown in Figure 3.

Figure 3

Solution

We know that $\frac{\pi }{6}$ is located in the first quadrant. However, $r=-2$. We can approach plotting a point with a negative $r$ in two ways:

1. Plot the point $\left(2,\frac{\pi }{6}\right)$ by moving $\frac{\pi }{6}$ in the counterclockwise direction and extending a directed line segment 2 units into the first quadrant. Then retrace the directed line segment back through the pole, and continue 2 units into the third quadrant;
2. Move $\frac{\pi }{6}$ in the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative direction, into the third quadrant.

See Figure 4(a). Compare this to the graph of the polar coordinate $\left(2,\frac{\pi }{6}\right)$ shown in Figure 4(b).

Figure 4

 Converting Between Polar Coordinates to Rectangular Coordinates

Figure 5

Solution

Use the equivalent relationships.

$\begin{array}{l}\begin{array}{l}\\ x=r\cos \theta \end{array}\hfill \\ x=3\cos \frac{\pi }{2}=0\hfill \\ y=r\sin \theta \hfill \\ y=3\sin \frac{\pi }{2}=3\hfill \end{array}$

The rectangular coordinates are $\left(0,3\right)$.

Figure 6

Solution

See Figure 7. Writing the polar coordinates as rectangular, we have

$\begin{array}{l}x=r\cos \theta \hfill \\ x=-2\cos \left(0\right)=-2\hfill \\ \hfill \\ y=r\sin \theta \hfill \\ y=-2\sin \left(0\right)=0\hfill \end{array}$

The rectangular coordinates are also $\left(-2,0\right)$.

Figure 7

Converting from Rectangular Coordinates to Polar Coordinates

A General Note: Converting from Rectangular Coordinates to Polar Coordinates

Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated in Figure 8.

$\begin{array}{l}\cos \theta =\frac{x}{r}\text{ or}x=r\cos \theta \hfill \\ \sin \theta =\frac{y}{r}\text{ or}y=r\sin \theta \hfill \\ {r}^{2}={x}^{2}+{y}^{2}\hfill \\ \tan \theta =\frac{y}{x}\hfill \end{array}$

Figure 8

Solution

We see that the original point $\left(3,3\right)$ is in the first quadrant. To find $\theta$, use the formula $\tan \theta =\frac{y}{x}$. This gives

$\begin{array}{l}\tan \theta =\frac{3}{3}\hfill \\ \tan \theta =1\hfill \\ {\tan }^{-1}\left(1\right)=\frac{\pi }{4}\hfill \end{array}$

To find $r$, we substitute the values for $x$ and $y$ into the formula $r=\sqrt{{x}^{2}+{y}^{2}}$. We know that $r$ must be positive, as $\frac{\pi }{4}$ is in the first quadrant. Thus

$\begin{array}{l}\begin{array}{l}\\ r=\sqrt{{3}^{2}+{3}^{2}}\end{array}\hfill \\ r=\sqrt{9+9}\hfill \\ r=\sqrt{18}=3\sqrt{2}\hfill \end{array}$

So, $r=3\sqrt{2}$ and $\theta \text{=}\frac{\pi }{4}$, giving us the polar point $\left(3\sqrt{2},\frac{\pi }{4}\right)$.

Figure 9

Analysis of the Solution

There are other sets of polar coordinates that will be the same as our first solution. For example, the points $\left(-3\sqrt{2},\frac{5\pi }{4}\right)$ and $\left(3\sqrt{2},-\frac{7\pi }{4}\right)$ will coincide with the original solution of $\left(3\sqrt{2},\frac{\pi }{4}\right)$. The point $\left(-3\sqrt{2},\frac{5\pi }{4}\right)$ indicates a move further counterclockwise by $\pi$, which is directly opposite $\frac{\pi }{4}$. The radius is expressed as $-3\sqrt{2}$. However, the angle $\frac{5\pi }{4}$ is located in the third quadrant and, as $r$ is negative, we extend the directed line segment in the opposite direction, into the first quadrant. This is the same point as $\left(3\sqrt{2},\frac{\pi }{4}\right)$. The point $\left(3\sqrt{2},-\frac{7\pi }{4}\right)$ is a move further clockwise by $-\frac{7\pi }{4}$, from $\frac{\pi }{4}$. The radius, $3\sqrt{2}$, is the same.

Transforming Equations between Polar and Rectangular Forms

We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.

How To: Given an equation in polar form, graph it using a graphing calculator.

1. Change the MODE to POL, representing polar form.
2. Press the Y= button to bring up a screen allowing the input of six equations: ${r}_{1},{r}_{2},...,{r}_{6}$.
3. Enter the polar equation, set equal to $r$.
4. Press GRAPH.

Example 6: Writing a Cartesian Equation in Polar Form

Write the Cartesian equation ${x}^{2}+{y}^{2}=9$ in polar form.

Solution

The goal is to eliminate $x$ and $y$ from the equation and introduce $r$ and $\theta$. Ideally, we would write the equation $r$ as a function of $\theta$. To obtain the polar form, we will use the relationships between $\left(x,y\right)$ and $\left(r,\theta \right)$. Since $x=r\cos \theta$ and $y=r\sin \theta$, we can substitute and solve for $r$.

$\begin{array}{ll}\text{ }{\left(r\cos \theta \right)}^{2}+{\left(r\sin \theta \right)}^{2}=9\hfill & \hfill \\ \text{ }{r}^{2}{\cos }^{2}\theta +{r}^{2}{\sin }^{2}\theta =9\hfill & \hfill \\ \text{ }{r}^{2}\left({\cos }^{2}\theta +{\sin }^{2}\theta \right)=9\hfill & \hfill \\ \text{ }{r}^{2}\left(1\right)=9 \hfill & {\text{Substitute cos}}^{2}\theta +{\sin }^{2}\theta =1.\hfill \\ \text{ }r=\pm 3\begin{array}{cccc}& & & \end{array}\hfill & \text{Use the square root property}.\hfill \end{array}$

Figure 10. (a) Cartesian form ${x}^{2}+{y}^{2}=9$ (b) Polar form $r=3$

Thus, ${x}^{2}+{y}^{2}=9,r=3$, and $r=-3$ should generate the same graph.To graph a circle in rectangular form, we must first solve for $y$.

$\begin{array}{l}\begin{array}{l}\\ {x}^{2}+{y}^{2}=9\end{array}\hfill \\ \text{ }{y}^{2}=9-{x}^{2}\hfill \\ \text{ }y=\pm \sqrt{9-{x}^{2}}\hfill \end{array}$

Note that this is two separate functions, since a circle fails the vertical line test. Therefore, we need to enter the positive and negative square roots into the calculator separately, as two equations in the form ${Y}_{1}=\sqrt{9-{x}^{2}}$ and ${Y}_{2}=-\sqrt{9-{x}^{2}}$. Press GRAPH.

Example 7: Rewriting a Cartesian Equation as a Polar Equation

Rewrite the Cartesian equation ${x}^{2}+{y}^{2}=6y$ as a polar equation.

Solution

This equation appears similar to the previous example, but it requires different steps to convert the equation. We can still follow the same procedures we have already learned and make the following substitutions:

$\begin{array}{ll}{r}^{2}=6y\hfill & \text{Use }{x}^{2}+{y}^{2}={r}^{2}.\hfill \\ {r}^{2}=6r\sin \theta \hfill & \text{Substitute}y=r\sin \theta .\hfill \\ \text{ }{r}^{2}-6r\sin \theta =0\hfill & \text{Set equal to 0}.\hfill \\ \text{ }r\left(r - 6\sin \theta \right)=0\hfill & \text{Factor and solve}.\hfill \\ r=0\hfill & \text{We reject }r=0,\text{as it only represents one point, }\left(0,0\right).\hfill \\ \text{or}r=6\sin \theta \begin{array}{cccc}& & & \end{array}\hfill & \hfill \end{array}$

Therefore, the equations ${x}^{2}+{y}^{2}=6y$ and $r=6\sin \theta$ should give us the same graph.

Figure 11. (a) Cartesian form ${x}^{2}+{y}^{2}=6y$ (b) polar form $r=6\sin \theta$

The Cartesian or rectangular equation is plotted on the rectangular grid, and the polar equation is plotted on the polar grid. Clearly, the graphs are identical.

Example 8: Rewriting a Cartesian Equation in Polar Form

Rewrite the Cartesian equation $y=3x+2$ as a polar equation.

Solution

We will use the relationships $x=r\cos \theta$ and $y=r\sin \theta$.

$\begin{array}{ll}\text{ }y=3x+2\hfill & \hfill \\ \text{ }r\sin \theta =3r\cos \theta +2\hfill & \hfill \\ r\sin \theta -3r\cos \theta =2\hfill & \hfill \\ r\left(\sin \theta -3\cos \theta \right)=2\hfill & \text{Isolate }r.\hfill \\ \text{ }r=\frac{2}{\sin \theta -3\cos \theta }\begin{array}{cccc}& & & \end{array}\hfill & \text{Solve for }r.\hfill \end{array}$

Try It 4

Rewrite the Cartesian equation ${y}^{2}=3-{x}^{2}$ in polar form. Solution

 Identify and Graph Polar Equations by Converting to Rectangular Equations

Solution

The conversion is

$\begin{array}{l}\text{ }r=2\sec \theta \hfill \\ \text{ }r=\frac{2}{\cos \theta }\hfill \\ r\cos \theta =2\hfill \\ x=2\hfill \end{array}$

Notice that the equation $r=2\sec \theta$ drawn on the polar grid is clearly the same as the vertical line $x=2$ drawn on the rectangular grid. Just as $x=c$ is the standard form for a vertical line in rectangular form, $r=c\sec \theta$ is the standard form for a vertical line in polar form.

Figure 12. (a) Polar grid (b) Rectangular coordinate system

A similar discussion would demonstrate that the graph of the function $r=2\csc \theta$ will be the horizontal line $y=2$. In fact, $r=c\csc \theta$ is the standard form for a horizontal line in polar form, corresponding to the rectangular form $y=c$.

Solution

The goal is to eliminate $\theta$ and $r$, and introduce $x$ and $y$. We clear the fraction, and then use substitution. In order to replace $r$ with $x$ and $y$, we must use the expression ${x}^{2}+{y}^{2}={r}^{2}$.

$\begin{array}{llll}\text{ }r=\frac{3}{1 - 2\cos \theta }\hfill & \hfill & \hfill & \hfill \\ r\left(1 - 2\cos \theta \right)=3\hfill & \hfill & \hfill & \hfill \\ r\left(1 - 2\left(\frac{x}{r}\right)\right)=3\hfill & \hfill & \hfill & \text{Use }\cos \theta =\frac{x}{r}\text{ to eliminate }\theta .\hfill \\ \text{ }r - 2x=3\hfill & \hfill & \hfill & \hfill \\ \text{ }r=3+2x\hfill & \hfill & \hfill & \text{Isolate }r.\hfill \\ \text{ }{r}^{2}={\left(3+2x\right)}^{2}\hfill & \hfill & \hfill & \text{Square both sides}.\hfill \\ \text{ }{x}^{2}+{y}^{2}={\left(3+2x\right)}^{2}\hfill & \hfill & \hfill & \text{Use }{x}^{2}+{y}^{2}={r}^{2}.\hfill \end{array}$

The Cartesian equation is ${x}^{2}+{y}^{2}={\left(3+2x\right)}^{2}$. However, to graph it, especially using a graphing calculator or computer program, we want to isolate $y$.

$\begin{array}{l}{x}^{2}+{y}^{2}={\left(3+2x\right)}^{2}\hfill \\ \text{ }{y}^{2}={\left(3+2x\right)}^{2}-{x}^{2}\hfill \\ \text{ }y=\pm \sqrt{{\left(3+2x\right)}^{2}-{x}^{2}}\hfill \end{array}$

When our entire equation has been changed from $r$ and $\theta$ to $x$ and $y$, we can stop, unless asked to solve for $y$ or simplify.

Figure 13

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.

Key Equations

Key Concepts

• The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.
• To plot a point in the form $\left(r,\theta \right),\theta >0$, move in a counterclockwise direction from the polar axis by an angle of $\theta$, and then extend a directed line segment from the pole the length of $r$ in the direction of $\theta$. If $\theta$ is negative, move in a clockwise direction, and extend a directed line segment the length of $r$ in the direction of $\theta$.
• If $r$ is negative, extend the directed line segment in the opposite direction of $\theta$.
• To convert from polar coordinates to rectangular coordinates, use the formulas $x=r\cos \theta$ and $y=r\sin \theta$.
• To convert from rectangular coordinates to polar coordinates, use one or more of the formulas: $\cos \theta =\frac{x}{r},\sin \theta =\frac{y}{r},\tan \theta =\frac{y}{x}$, and $r=\sqrt{{x}^{2}+{y}^{2}}$.
• Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations.
• Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and then graph it in the rectangular plane.

Glossary

polar axis

on the polar grid, the equivalent of the positive x-axis on the rectangular grid

polar coordinates

on the polar grid, the coordinates of a point labeled $\left(r,\theta \right)$, where $\theta$ indicates the angle of rotation from the polar axis and $r$ represents the radius, or the distance of the point from the pole in the direction of $\theta$

pole

the origin of the polar grid

Section Exercises

1. How are polar coordinates different from rectangular coordinates? 2. How are the polar axes different from the x- and y-axes of the Cartesian plane? 3. Explain how polar coordinates are graphed. 4. How are the points $\left(3,\frac{\pi }{2}\right)$ and $\left(-3,\frac{\pi }{2}\right)$ related? 5. Explain why the points $\left(-3,\frac{\pi }{2}\right)$ and $\left(3,-\frac{\pi }{2}\right)$ are the same. For the following exercises, convert the given polar coordinates to Cartesian coordinates with $r>0$ and $0\le \theta \le 2\pi$. Remember to consider the quadrant in which the given point is located when determining $\theta$ for the point. 6. $\left(7,\frac{7\pi }{6}\right)$ 7. $\left(5,\pi \right)$ 8. $\left(6,-\frac{\pi }{4}\right)$ 9. $\left(-3,\frac{\pi }{6}\right)$ 10. $\left(4,\frac{7\pi }{4}\right)$ For the following exercises, convert the given Cartesian coordinates to polar coordinates with $r>0,0\le \theta <2\pi$. Remember to consider the quadrant in which the given point is located. 11. $\left(4,2\right)$ 12. $\left(-4,6\right)$ 13. $\left(3,-5\right)$ 14. $\left(-10,-13\right)$ 15. $\left(8,8\right)$ For the following exercises, convert the given Cartesian equation to a polar equation. 16. $x=3$ 17. $y=4$ 18. $y=4{x}^{2}$ 19. $y=2{x}^{4}$ 20. ${x}^{2}+{y}^{2}=4y$ 21. ${x}^{2}+{y}^{2}=3x$ 22. ${x}^{2}-{y}^{2}=x$ 23. ${x}^{2}-{y}^{2}=3y$ 24. ${x}^{2}+{y}^{2}=9$ 25. ${x}^{2}=9y$ 26. ${y}^{2}=9x$ 27. $9xy=1$ For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented. 28. $r=3\sin \theta$ 29. $r=4\cos \theta$ 30. $r=\frac{4}{\sin \theta +7\cos \theta }$ 31. $r=\frac{6}{\cos \theta +3\sin \theta }$ 32. $r=2\sec \theta$ 33. $r=3\csc \theta$ 34. $r=\sqrt{r\cos \theta +2}$ 35. ${r}^{2}=4\sec \theta \csc \theta$ 36. $r=4$ 37. ${r}^{2}=4$ 38. $r=\frac{1}{4\cos \theta -3\sin \theta }$ 39. $r=\frac{3}{\cos \theta -5\sin \theta }$ For the following exercises, find the polar coordinates of the point. 40. 41. 42. 43. 44. For the following exercises, plot the points. 45. $\left(-2,\frac{\pi }{3}\right)$ 46. $\left(-1,-\frac{\pi }{2}\right)$ 47. $\left(3.5,\frac{7\pi }{4}\right)$ 48. $\left(-4,\frac{\pi }{3}\right)$ 49. $\left(5,\frac{\pi }{2}\right)$ 50. $\left(4,\frac{-5\pi }{4}\right)$ 51. $\left(3,\frac{5\pi }{6}\right)$ 52. $\left(-1.5,\frac{7\pi }{6}\right)$ 53. $\left(-2,\frac{\pi }{4}\right)$ 54. $\left(1,\frac{3\pi }{2}\right)$ For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis. 55. $5x-y=6$ 56. $2x+7y=-3$ 57. ${x}^{2}+{\left(y - 1\right)}^{2}=1$ 58. ${\left(x+2\right)}^{2}+{\left(y+3\right)}^{2}=13$ 59. $x=2$ 60. ${x}^{2}+{y}^{2}=5y$ 61. ${x}^{2}+{y}^{2}=3x$ For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane. 62. $r=6$ 63. $r=-4$ 64. $\theta =-\frac{2\pi }{3}$ 65. $\theta =\frac{\pi }{4}$ 66. $r=\sec \theta$ 67. $r=-10\sin \theta$ 68. $r=3\cos \theta$ 69. Use a graphing calculator to find the rectangular coordinates of $\left(2,-\frac{\pi }{5}\right)$. Round to the nearest thousandth. 70. Use a graphing calculator to find the rectangular coordinates of $\left(-3,\frac{3\pi }{7}\right)$. Round to the nearest thousandth. 71. Use a graphing calculator to find the polar coordinates of $\left(-7,8\right)$ in degrees. Round to the nearest thousandth. 72. Use a graphing calculator to find the polar coordinates of $\left(3,-4\right)$ in degrees. Round to the nearest hundredth. 73. Use a graphing calculator to find the polar coordinates of $\left(-2,0\right)$ in radians. Round to the nearest hundredth. 74. Describe the graph of $r=a\sec \theta ;a>0$. 75. Describe the graph of $r=a\sec \theta ;a<0$. 76. Describe the graph of $r=a\csc \theta ;a>0$. 77. Describe the graph of $r=a\csc \theta ;a<0$. 78. What polar equations will give an oblique line? For the following exercises, graph the polar inequality. 79. $r<4$ 80. $0\le \theta \le \frac{\pi }{4}$ 81. $\theta =\frac{\pi }{4},r\ge 2$ 82. $\theta =\frac{\pi }{4},r\ge -3$ 83. $0\le \theta \le \frac{\pi }{3},r<2$ 84. $\frac{-\pi }{6}<\theta \le \frac{\pi }{3},-3<r<2$

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