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Chapter 8: Problem 64

Sketch a graph of the polar equation and find the tangents at the pole. $$ r=-\sin 5 \theta $$

Short Answer

Expert verified
The tangents at the pole can be drawn at points corresponding to the angles where the sign of \(r\) changes from negative to positive or vice-versa. The graph of the polar equation consists of a series of loops: five petals pointing in opposite directions.

Step by step solution

01

Sketching the graph

To start, plot the graph of the polar equation \(r = -\sin 5 \theta\). To do this, identify some key points by plugging in values for \(\theta\) into the equation and solving for \(r\). This will help create a table of values to use as a guide while sketching the graph.
02

Interpreting the negative radius

In polar coordinates, if the radius \(r\) is negative, it simply means that the point lies in the opposite direction. After plotting the positive radii, mirror-image those points across the pole to account for the negative radii depicted in the polar equation.
03

Tracing the curve

Once all key points are plotted, trace the curve using those points. The spikes of the graph occur at the origin; curve goes to the origin as \(\theta\) increases in increments of \(\frac{\pi}{5}\).
04

Identifying locations for the tangents at the pole

Tangents to the polar graph at the pole occur when \(r\) changes from negative to positive or vice-versa. Identify the values of \(\theta\) where \(r\) changes sign. Then, draw the tangents at these points.

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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. $$ x=1+\frac{1}{t}, \quad y=t-1 $$

True or False. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graph of the parametric equations \(x=t^{2}\) and \(y=t^{2}\) is the line \(y=x\).

Find the area of the circle given by \(r=\sin \theta+\cos \theta\). Check your result by converting the polar equation to rectangular form, then using the formula for the area of a circle.

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{2}{2+3 \sin \theta}\)

Find two different sets of parametric equations for the rectangular equation. $$ y=3 x-2 $$
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