7.7: Polar Coordinates
- Page ID
- 155877
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The position of a point in the plane can be described by its distance and direction from the origin. In measuring direction we take the \(x\)-axis as the starting point. Let \(X\) be the point \((1,0)\) on the \(x\)-axis and let \(P\) be a point in the plane as in Figure 7.7.1.
A pair of polar coordinates of \(P\) is given by \((r, \theta)\) where \(r\) is the distance from the origin to \(P\) and \(\theta\) is the angle \(X O P\).
Each pair of real numbers \((r, \theta)\) determines a point \(P\) in polar coordinates. To find \(P\) we first rotate the line \(O X\) through an angle \(\theta\), forming a new line \(O X^{\prime}\), and then go out a distance \(r\) along the line \(O X^{\prime}\). If \(\theta\) is negative then the rotation is in the negative, or clockwise direction. If \(r\) is negative the distance is measured along the line \(O X^{\prime}\) in the direction away from \(X^{\prime}\) (see Figure 7.7.2).
Plot the following points in polar coordinates.
Solution
\[(2, \pi / 4), \quad(-1, \pi / 4), \quad(3,3 \pi / 4), \quad(2,-\pi / 4), \quad(-4,-\pi / 4) \nonumber \]
The solution is shown in Figure 7.7.3.
Each point \(P\) has infinitely many different polar coordinate pairs. We see in Figure 7.7.4 that the point \(P(3, \pi / 2)\) has all the coordinates
\[\left.\begin{array}{l} (3, \pi / 2+2 n \pi) \\ (-3,3 \pi / 2+2 n \pi) \end{array}\right\} n \text { an integer. } \nonumber \]
Figure 7.7.4
Any coordinate pair \((0, \theta)\) with \(r=0\) determines the origin. As we see in Figure 7.7.5, the coordinates of a point \(P\) in rectangular and in polar coordinates are related by the equations
\[x=r \cos \theta, \quad y=r \sin \theta \nonumber \]
The graph, or locus in polar coordinates of a system of formulas in the variables \(r, \theta\) is the set of all points \(P(r, \theta)\) for which the formulas are true.
The graph of the equation \(r=a\) is the circle of radius \(a\) centered at the origin (Figure 7.7.6(a)). The graph of the equation \(\theta=b\) is a straight line through the origin (Figure 7.7.6(b)).
Solution
EXAMPLE 3 The graph of the system of formulas
\[r=\theta, \quad 0 \leq \theta \nonumber \]
is the spiral of Archimedes formed by moving a pencil along the line \(O X\) while the line is rotating, with the pencil moving at the same speed as the point \(X\). The graph is shown in Figure 7.7.6(c).
An equation in rectangular coordinates can readily be transformed into an equation in polar coordinates with the same graph by using \(x=r \cos \theta, y=r \sin 0\).
(a)
(b)
(c)
Here are the polar equations for various types of straight lines. Examples of their graphs are shown in Figure 7.7.7.
(1) Line through the origin (not vertical).
Rectangular equation: \(y=m x\).
Polar equation: \(\quad r \sin \theta=m r \cos \theta\),
or:
\(\tan \theta=m\)
(2) Horizontal line (not through origin).
Rectangular equation: \(\quad y=b\).
Polar equation: \(\quad r \sin \theta=b\),
or:
\[r=b \csc \theta \nonumber \]
(3) Vertical line (not through origin).
Rectangular equation: \(\quad x=a\).
Polar equation: \(\quad r \cos \theta=a\),
or:
\[r=a \sec \theta \nonumber \]
(4) Vertical line through origin.
Rectangular equation: \(\quad x=0\).
Polar equation: \(\quad r \cos \theta=0\),
or:
\[\theta=\pi / 2 \nonumber \]
(5) Other lines.
Rectangular equation: \(\quad y=m x+b\).
Polar equation:
\[r \sin \theta=m r \cos \theta+b \nonumber \]
or:
\[r=\frac{b}{\sin \theta-m \cos \theta} \nonumber \]
\(y=m x, \tan \theta=m\)
\(x=a, r=a \sec \theta\)
\(y=b, r=b \csc \theta\)
\[y=m x+b, r=\frac{b}{\sin \theta-m \cos \theta} \nonumber \]
The parabola \(y=x^{2}\) has the polar equation
Solution
\[r \sin \theta=(r \cos \theta)^{2}, \text { or } \quad r=\frac{\sin \theta}{\cos ^{2} \theta}=\tan \theta \sec \theta \nonumber \]
The curve \(y=1 / x\) has the polar equation
Solution
\[r \sin \theta=\frac{1}{r \cos \theta}, \quad \text { or } \quad r^{2}=\sec \theta \csc \theta \nonumber \]
The graph is shown in Figure 7.7.8.
Some curves have much simpler equations in polar coordinates than in rectangular coordinates.
The graph of the equation
Solution
\[r=a \sin \theta \nonumber \]
is the circle one of whose diameters is the line from the origin to a point \(a\) above the origin.
This can be seen from Figure 7.7.9, if we remember that a diameter and a point on the circle form a right triangle.
As \(\theta\) increases, the point ( \(a \sin \theta, \theta\) ) goes around this circle once for every \(\pi\) radians.
Figure 7.7.10
An equation \(r=f(\theta)\) in polar coordinates has the same graph as the pair of parametric equations
\[x=f(\theta) \cos \theta, \quad y=f(\theta) \sin \theta \nonumber \]
in rectangular coordinates. This can be seen from Figure 7.7.10.
EXAMPLE 7
(a) The spiral \(r=\theta\) has the parametric equations
\[x=\theta \cos \theta, \quad y=\theta \sin \theta \nonumber \]
(b) The circle \(r=a \sin \theta\) has the parametric equations
\[x=a \sin \theta \cos \theta, \quad y=a \sin ^{2} \theta \nonumber \]
PROBLEMS FOR SECTION 7.7
1 Plot the following points in polar coordinates:
(a) \((2, \pi / 3)\)
(b) \((-3, \pi / 2)\)
(c) \((1,4 \pi / 3)\)
(d) \((-2,-\pi / 4)\)
(e) \(\left(\frac{1}{2}, \pi\right)\)
(f) \((0,3 \pi / 2)\)
In Problems 2-12, find an equation in polar coordinates which has the same graph as the given equation in rectangular coordinates.
\[\begin{array}{rlrl} \mathbf{2} & y=3 x & \mathbf{3} & y=5 x+2 \\ \mathbf{4} & y=-4 & \mathbf{5} & x=2 \\ \mathbf{6} & x y^{2}=1 & \mathbf{7} & y=x^{2}+1 \\ \mathbf{8} & x^{2}+y^{2}=5 & \mathbf{9} & y=3 x^{2}-2 x \\ \mathbf{1 0} & y=x^{3} & \mathbf{1 1} & y=x^{2}+y^{2} \\ \mathbf{1 2} & y=\sin x & & \end{array} \nonumber \]
In Problems 13-20, sketch the given curve in polar coordinates.
13
\(r=\cos \theta\)
\(14 r=-\sec \theta\)
\(15 r=\sin (\theta+\pi / 4)\)
\(16 \quad r=0, \quad 0 \leq 0\)
\[\begin{array}{llll} 17 & r=1+\theta^{2} / \pi^{2} & 18 & r=\frac{1}{\sin \theta+\cos \theta} \\ 19 & r=\cot \theta \csc \theta & 20 & r^{2}=-2 \sec \theta \csc \theta \end{array} \nonumber \]
In Problems 21-24, find rectangular parametric equations for the given curves.
| 21 | \(r=\sin (3 \theta)\) | 22 | \(r=\sec \theta \csc \theta\) |
| 23 | \(r=\theta^{2}\) | 24 | \(r=\tan \theta\) |
25 Prove that if \(f(\theta)=f(-\theta)\) then the curve \(r=f(\theta)\) is symmetric about the \(x\)-axis. That is, if \((x, y)\) is on the curve then so is \((x,-y)\).
26 Prove that if \(f(\theta)=f(\pi+\theta)\) then the curve \(r=f(\theta)\) is symmetric about the origin. That is, if \((x, y)\) is on the curve so is \((-x,-y)\)
27 Prove that if \(f(\theta)=f(\pi-\theta)\) then the curve \(r=f(\theta)\) is symmetric about the \(y\)-axis