/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 72 Sketch a graph of the polar equa... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 72

Sketch a graph of the polar equation. $$ r=\frac{6}{2 \sin \theta-3 \cos \theta} $$

Short Answer

Expert verified
The graph of the equation \(r=\frac{6}{2 \sin \theta-3 \cos \theta}\) can be sketched by choosing a range of 胃 values, calculating the corresponding r values, plotting these points on a polar grid, and connecting them to visualize the graph. Look out for places where the function is undefined.

Step by step solution

01

Choose values for theta

Start by choosing a range of values for 胃. A complete range is usually from 0 to 2蟺 (0 to 360 degrees), as this covers a complete circle. You can pick more specific values if needed, but a basic set to start with is every 30 or 45 degrees.
02

Calculate the values of r

For each chosen value of 胃, use the polar equation to find the corresponding r values. Substitute 胃 into the equation \(r=\frac{6}{2 \sin \theta-3 \cos \theta}\) and calculate the values.
03

Plot the values

Plot the values you calculated on a polar grid. Remember that in polar coordinates, the angle 胃 is the counterclockwise angle from the x-axis, and r is the distance from the origin. Connect the points to visualize the graph of the equation.
04

Check for Undefined Values

Take note where the denominator \(2 \sin \theta - 3 \cos \theta\) equals zero, as these values for 胃 will make the function undefined. You should mark these angles on your graph and determine the behavior of the graph near these points.

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Most popular questions from this chapter

In Exercises 43-46, find the area of the surface formed by revolving the curve about the given line. $$ \begin{array}{lll} \underline{\text { Polar Equation }} & \underline{\text { Interval }} & \underline{\text { Axis of Revolution }} \\ r=a \cos \theta & 0 \leq \theta \leq \frac{\pi}{2} & \theta=\frac{\pi}{2} \end{array} $$

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 \(7-16,\) 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. \(r=\frac{-6}{3+7 \sin \theta}\)

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

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Witch of Agnesi: } x=2 \cot \theta, \quad y=2 \sin ^{2} \theta $$
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