Slope Of Polar Curve Formula

Learning Outcomes

Sketch polar curves from given equations
Convert equations betwixt rectangular and polar coordinates
Place symmetry in polar curves and equations

Polar Curves

Now that nosotros know how to plot points in the polar coordinate system, we can discuss how to plot curves. In the rectangular coordinate system, we can graph a function [latex]y=f\left(x\right)[/latex] and create a curve in the Cartesian plane. In a similar fashion, nosotros can graph a curve that is generated by a function [latex]r=f\left(\theta \right)[/latex].

The general idea behind graphing a function in polar coordinates is the same equally graphing a function in rectangular coordinates. Start with a list of values for the independent variable ([latex]\theta[/latex] in this instance) and calculate the corresponding values of the dependent variable [latex]r[/latex]. This process generates a list of ordered pairs, which tin can exist plotted in the polar coordinate arrangement. Finally, connect the points, and take advantage of any patterns that may appear. The role may be periodic, for example, which indicates that merely a limited number of values for the independent variable are needed.

Problem-Solving Strategy: Plotting a Curve in Polar Coordinates

Create a table with ii columns. The first column is for [latex]\theta [/latex], and the second column is for [latex]r[/latex].
Create a list of values for [latex]\theta [/latex].
Calculate the respective [latex]r[/latex] values for each [latex]\theta [/latex].
Plot each ordered pair [latex]\left(r,\theta \right)[/latex] on the coordinate axes.
Connect the points and look for a pattern.

Example: Graphing a Function in Polar Coordinates

Graph the curve defined by the part [latex]r=4\sin\theta [/latex]. Identify the curve and rewrite the equation in rectangular coordinates.

Lookout the post-obit video to see the worked solution to Example: Graphing a Function in Polar Coordinates.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point every bit this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of "vii.three Polar Coordinates" here (opens in new window).

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Create a graph of the curve defined by the part [latex]r=4+4\cos\theta [/latex].

The graph in the previous example was that of a circumvolve. The equation of the circle can be transformed into rectangular coordinates using the coordinate transformation formulas in the theorem. The example later on the side by side gives some more than examples of functions for transforming from polar to rectangular coordinates.

Example: Transforming Polar Equations to Rectangular Coordinates

Rewrite each of the following equations in rectangular coordinates and identify the graph.

[latex]\theta =\frac{\pi }{3}[/latex]
[latex]r=3[/latex]
[latex]r=6\cos\theta -eight\sin\theta [/latex]

Watch the following video to meet the worked solution to Example: Transforming Polar Equations to Rectangular Coordinates.

For airtight captioning, open up the video on its original page by clicking the Youtube logo in the lower right-paw corner of the video brandish. In YouTube, the video will brainstorm at the same starting indicate as this clip, simply volition continue playing until the very stop.

You lot tin can view the transcript for this segmented clip of "seven.iii Polar Coordinates" here (opens in new window).

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Rewrite the equation [latex]r=\sec\theta \tan\theta [/latex] in rectangular coordinates and identify its graph.

Nosotros have now seen several examples of drawing graphs of curves defined by polar equations. A summary of some mutual curves is given in the tables below. In each equation, a and b are arbitrary constants.

This table has three columns and 3 rows. The first row is a header row and is given from left to right as name, equation, and example. The second row is Line passing through the pole with slope tan K; θ = K; and a picture of a straight line on the polar coordinate plane with θ = π/3. The third row is Circle; r = a cosθ + b sinθ; and a picture of a circle on the polar coordinate plane with equation r = 2 cos(t) – 3 sin(t): the circle touches the origin but has center in the third quadrant.

This table has three columns and 3 rows. The first row is Spiral; r = a + bθ; and a picture of a spiral starting at the origin with equation r = θ/3. The second row is Cardioid; r = a(1 + cosθ), r = a(1 – cosθ), r = a(1 + sinθ), r = a(1 – sinθ); and a picture of a cardioid with equation r = 3(1 + cosθ): the cardioid looks like a heart turned on its side with a rounded bottom instead of a pointed one. The third row is Limaçon; r = a cosθ + b, r = a sinθ + b; and a picture of a limaçon with equation r = 2 + 4 sinθ: the figure looks like a deformed circle with a loop inside of it. The seventh row is Rose; r = a cos(bθ), r = a sin(bθ); and a picture of a rose with equation r = 3 sin(2θ): the rose looks like a flower with four petals, one petal in each quadrant, each with length 3 and reaching to the origin between each petal.

A cardioid is a special case of a limaçon (pronounced "lee-mah-son"), in which [latex]a=b[/latex] or [latex]a=-b[/latex]. The rose is a very interesting bend. Notice that the graph of [latex]r=3\sin2\theta [/latex] has four petals. However, the graph of [latex]r=three\sin3\theta [/latex] has 3 petals equally shown.

A rose with three petals, one in the first quadrant, another in the second quadrant, and the third in both the third and fourth quadrants, each with length 3. Each petal starts and ends at the origin.

If the coefficient of [latex]\theta [/latex] is even, the graph has twice as many petals every bit the coefficient. If the coefficient of [latex]\theta [/latex] is odd, and then the number of petals equals the coefficient. You are encouraged to explore why this happens. Even more than interesting graphs emerge when the coefficient of [latex]\theta [/latex] is not an integer. For example, if it is rational, then the curve is closed; that is, information technology eventually ends where information technology started (Figure ten (a)). However, if the coefficient is irrational, so the curve never closes (Figure x (b)). Although information technology may announced that the curve is airtight, a closer exam reveals that the petals but above the positive x centrality are slightly thicker. This is because the petal does not quite match up with the starting point.

This figure has two figures. The first is a rose with so many overlapping petals that there are a few patterns that develop, starting with a sharp 10 pointed star in the center and moving out to an increasingly rounded set of petals. The second figure is a rose with even more overlapping petals, so many so that it is impossible to tell what is happening in the center, but on the outer edges are a number of sharply rounded petals.

Since the curve defined by the graph of [latex]r=3\sin\left(\pi \theta \right)[/latex] never closes, the bend depicted in Effigy ten (b) is just a partial depiction. In fact, this is an instance of a space-filling curve. A space-filling curve is one that in fact occupies a two-dimensional subset of the real aeroplane. In this case the curve occupies the circumvolve of radius three centered at the origin.

Suppose a bend is described in the polar coordinate system via the function [latex]r=f\left(\theta \right)[/latex]. Since we have conversion formulas from polar to rectangular coordinates given by

[latex]\begin{array}{c}x=r\cos\theta \hfill \\ y=r\sin\theta ,\hfill \end{array}[/latex]

it is possible to rewrite these formulas using the function

[latex]\begin{array}{c}x=f\left(\theta \right)\cos\theta \hfill \\ y=f\left(\theta \correct)\sin\theta .\hfill \finish{array}[/latex]

This stride gives a parameterization of the curve in rectangular coordinates using [latex]\theta [/latex] as the parameter. For example, the spiral formula [latex]r=a+b\theta [/latex] from Figure seven becomes

[latex]\begin{assortment}{c}x=\left(a+b\theta \right)\cos\theta \hfill \\ y=\left(a+b\theta \right)\sin\theta .\hfill \stop{assortment}[/latex]

Letting [latex]\theta [/latex] range from [latex]-\infty [/latex] to [latex]\infty [/latex] generates the entire spiral.

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Symmetry in Polar Coordinates

When studying symmetry of functions in rectangular coordinates (i.due east., in the class [latex]y=f\left(x\right)[/latex]), we talk about symmetry with respect to the y-axis and symmetry with respect to the origin. In detail, if [latex]f\left(-ten\right)=f\left(x\right)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex], then [latex]f[/latex] is an even function and its graph is symmetric with respect to the [latex]y[/latex]-centrality. If [latex]f\left(-10\correct)=-f\left(x\right)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex], then [latex]f[/latex] is an odd function and its graph is symmetric with respect to the origin. By determining which types of symmetry a graph exhibits, we can learn more about the shape and appearance of the graph. Symmetry can also reveal other properties of the function that generates the graph. Symmetry in polar curves works in a similar mode.

theorem: Symmetry in Polar Curves and Equations

Consider a bend generated by the function [latex]r=f\left(\theta \right)[/latex] in polar coordinates.

The curve is symmetric about the polar axis if for every point [latex]\left(r,\theta \right)[/latex] on the graph, the indicate [latex]\left(r,-\theta \right)[/latex] is besides on the graph. Similarly, the equation [latex]r=f\left(\theta \correct)[/latex] is unchanged by replacing [latex]\theta [/latex] with [latex]-\theta [/latex].
The bend is symmetric most the pole if for every point [latex]\left(r,\theta \correct)[/latex] on the graph, the point [latex]\left(r,\pi +\theta \right)[/latex] is also on the graph. Similarly, the equation [latex]r=f\left(\theta \right)[/latex] is unchanged when replacing [latex]r[/latex] with [latex]-r[/latex], or [latex]\theta [/latex] with [latex]\pi +\theta [/latex].
The bend is symmetric nigh the vertical line [latex]\theta =\frac{\pi }{2}[/latex] if for every signal [latex]\left(r,\theta \correct)[/latex] on the graph, the point [latex]\left(r,\pi -\theta \right)[/latex] is likewise on the graph. Similarly, the equation [latex]r=f\left(\theta \right)[/latex] is unchanged when [latex]\theta [/latex] is replaced by [latex]\pi -\theta [/latex].

The following table shows examples of each blazon of symmetry.

This table has three rows and two columns. The first row reads

Effigy 11.

Case: using Symmetry to Graph a Polar Equation

Find the symmetry of the rose divers past the equation [latex]r=3\sin\left(ii\theta \right)[/latex] and create a graph.

Lookout the following video to see the worked solution to Example: using Symmetry to Graph a Polar Equation.

For closed captioning, open up the video on its original folio past clicking the Youtube logo in the lower right-mitt corner of the video display. In YouTube, the video will brainstorm at the same starting point as this clip, but will keep playing until the very stop.

You can view the transcript for this segmented prune of "7.3 Polar Coordinates" here (opens in new window).

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Determine the symmetry of the graph determined by the equation [latex]r=2\cos\left(3\theta \right)[/latex] and create a graph.

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