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Chapter 8: Problem 68

Sketch a graph of the polar equation. $$ r=1+\sin \theta $$

Short Answer

Expert verified
To sketch the graph of the polar equation \(r = 1+\sin\theta\), you have to plot the curve by evaluating \(r\) at various values of \(\theta\) within the range \(0 \leq \theta \leq 2\pi\). The resulting plot reveals a heart-shaped curve known as a cardioid.

Step by step solution

01

Set up the range of theta

Start by determining the values of theta within the standard range for trigonometric functions: \(0 \leq \theta \leq 2\pi\). You may want to use intervals of \(\frac{\pi}{2}\) for convenience as all usable values of the sine function are obtained within this range.
02

Evaluate r

Calculate the corresponding values of \(r\) by substituting the chosen values of theta into the equation \(r = 1+\sin{\theta}\). This step gives us the distance from the origin of each point on the curve.
03

Plot the curve

Now is the time to plot the curve. Each point on the curve is represented by a direction and a distance from the origin (0,0) in polar coordinates. Use the values from steps 1 and 2 to plot these points, and then connect these points smoothly to form the curve. You'll notice the unique pattern of this curve, which is sometimes called a 'cardioid' due to its heart-like shape.

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Most popular questions from this chapter

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 t, \quad y=|t-2| $$

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \text { Circle: } x=h+r \cos \theta, \quad y=k+r \sin \theta $$

A curve called the folium of Descartes can be represented by the parametric equations \(x=\frac{3 t}{1+t^{3}} \quad\) and \(y=\frac{3 t^{2}}{1+t^{3}}\) (a) Convert the parametric equations to polar form. (b) Sketch the graph of the polar equation from part (a). (c) Use a graphing utility to approximate the area enclosed by the loop of the curve.

Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? $$ \text { (a) } \begin{aligned} x &=t \\ y &=2 t+1 \end{aligned} $$ $$ \text { (b) } \begin{aligned} x &=\cos \theta \\ y &=2 \cos \theta+1 \end{aligned} $$ $$ \text { (c) } \begin{aligned} x &=e^{-t} \\ y &=2 e^{-t}+1 \end{aligned} $$ (d) \(x=e^{t}\) $$ y=2 e^{t}+1 $$

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r(3-2 \cos \theta)=6\)
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