[latex]{r}_{1}=1+\sin \theta ,{r}_{2}=3\sin \theta [/latex]67. [latex]{r}_{1}=6 - 4\cos \theta ,{r}_{2}=4[/latex]66. [latex]r=\theta ,r=\theta +\sin \theta [/latex]56. [latex]{r}^{2}=10\cos \left(2\theta \right)[/latex]34. For [latex]r=a\sin \theta [/latex], the center is [latex]\left(\frac{a}{2},\pi \right)[/latex]. Figures 20 and 21 summarize the graphs and equations for each of these curves.1.
Explain the similarities and differences you observe in the graphs.For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.64. Look at the range of When these curves are drawn, it is best to plot the points in order, as in the table of Example 8’s solution. Explain the similarities and differences you observe in the graphs.63. [latex]r=4\text{sin}\left(4\theta \right)[/latex]40.
The graph is not only symmetric with respect to the polar axis, but also with respect to the line [latex]\theta =\frac{\pi }{2}[/latex] and the pole.Now we will find the zeros. Polar Equations & Graphs (Trigonometry/Precalculus) - YouTube
The zero of the equation is located at [latex]\left(0,\pi \right)[/latex].
We replace [latex]\left(r,\theta \right)[/latex] with [latex]\left(-r,\theta \right)[/latex] to determine if the tested equation is equivalent to the original equation. [latex]{r}^{2}=36\cos \left(2\theta \right)[/latex]33. [latex]{r}^{2}=4\sin \left(2\theta \right)[/latex]35. Consider [latex]r=5\cos \theta [/latex]; the maximum distance between the curve and the pole is 5 units. On a graphing utility, graph each polar equation. We use the same process for polar equations. Thus,The point [latex]\left(2,0\right)[/latex] is on the curve.The graph of the rose curve has unique properties, which are revealed in the table below.As [latex]r=0[/latex] when [latex]\theta =\frac{\pi }{8}[/latex], it makes sense to divide values in the table by [latex]\frac{\pi }{8}[/latex] units.
In the last two examples, the same equation was used to illustrate the properties of symmetry and demonstrate how to find the zeros, maximum values, and plotted points that produced the graphs. [latex]{r}_{1}=3+2\sin \theta ,{r}_{2}=2[/latex]65.
[latex]{r}_{1}=1+\cos \theta ,{r}_{2}=3\cos \theta [/latex]68.
Explain the similarities and differences you observe in the graphs.63. Figure 5 shows the graphs of these four circles.Sketch the graph of [latex]r=4\cos \theta [/latex].First, testing the equation for symmetry, we find that the graph is symmetric about the polar axis. The maximum value of the cosine function is 1 when [latex]\theta =0[/latex], so our polar equation is [latex]5\cos \theta [/latex], and the value [latex]\theta =0[/latex] will yield the maximum [latex]|r|[/latex].Similarly, the maximum value of the sine function is 1 when [latex]\theta =\frac{\pi }{2}[/latex], and if our polar equation is [latex]r=5\sin \theta [/latex], the value [latex]\theta =\frac{\pi }{2}[/latex] will yield the maximum [latex]|r|[/latex]. On a graphing utility, graph each polar equation. Set [latex]r=0[/latex], and solve for [latex]\theta [/latex].For many of the forms we will encounter, the maximum value of a polar equation is found by substituting those values of [latex]\theta [/latex] into the equation that result in the maximum value of the trigonometric functions. We can investigate other intercepts by calculating [latex]r[/latex] when [latex]\theta =0[/latex].So, there is at least one polar axis intercept at [latex]\left(4,0\right)[/latex].Next, as the maximum value of the sine function is 1 when [latex]\theta =\frac{\pi }{2}[/latex], we will substitute [latex]\theta =\frac{\pi }{2}[/latex] into the equation and solve for [latex]r[/latex]. To graph in the polar coordinate system we construct a table of [latex]\theta [/latex] and [latex]r[/latex] values.
[latex]{r}_{1}{}^{2}=\sin \theta ,{r}_{2}{}^{2}=\cos \theta [/latex]72.
Thus, [latex]r=1[/latex].Make a table of the coordinates similar to the table below.This is an example of a curve for which making a table of values is critical to producing an accurate graph. While it may be apparent that an equation involving [latex]\sin \theta [/latex] is likely symmetric with respect to the line [latex]\theta =\frac{\pi }{2}[/latex], evaluating more points helps to verify that the graph is correct.Sketch the graph of [latex]r=3 - 2\cos \theta [/latex].Sketch the graph of [latex]r=2+5\text{cos}\theta [/latex].Testing for symmetry, we find that the graph of the equation is symmetric about the polar axis. Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane.2. [latex]r=2\sin \theta \tan \theta [/latex], a cissoid47. On a graphing utility, graph and sketch [latex]r=\sin \theta +{\left(\sin \left(\frac{5}{2}\theta \right)\right)}^{3}[/latex] on [latex]\left[0,4\pi \right][/latex].61.
Further Applications of Trigonometry. [latex]r=4\text{sin}\left(5\theta \right)[/latex]For the following exercises, use a graphing calculator to sketch the graph of the polar equation.46. We may find additional information by calculating values of [latex]r[/latex] when [latex]\theta =0[/latex]. The angle [latex]\theta [/latex] is undefined for any value of [latex]\sin \theta >1[/latex]. We will want to make the substitution [latex]u=5\theta [/latex].The maximum value is calculated at the angle where [latex]\sin \theta [/latex] is a maximum.
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