/*! 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 61 Sketch a graph of the polar equa... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 61

Sketch a graph of the polar equation and find the tangents at the pole. $$ r=3 \sin \theta $$

Short Answer

Expert verified
The graph of our polar equation, \(r = 3 \sin \theta\), is a circle of radius 3 centered at the origin. This circle is traversed twice as \( \theta\) ranges from 0 to \( 2\pi\). The tangent lines at the pole, or origin, are the x-axis and negative x-axis.

Step by step solution

01

Understanding the Polar Equation

First, let's try to get a feel for our function. The given polar equation is \(r = 3 \sin \theta\). This describes a circle of radius 3 centered at the origin in the polar coordinate system. It is important to note that for \( \theta = 0 \) or \( \theta = \pi \), we have \( r = 0 \) or \( r = 3 \), respectively.
02

Sketching the Polar Graph

The nature of the sin function means that our circle is traversed as \(\theta\) ranges from 0 to \( \pi\), then backtracked (traced out again but in reverse order) as \(\theta\) ranges from \( \pi\) to \(2 \pi\). Therefore, we sketch a circle with radius 3 in the polar coordinate system. Please note that a negative value for r does not alter its direction but it changes the location of the point(pole).
03

Find the Tangents at the Pole

In general, if an equation of a curve in polar coordinates is given by \( r = f(\theta) \), then the polar tangents are given when \( dr/d\theta = 0 \). Differentiating \( r = 3 \sin \theta\) with respect to \(\theta\) gives \( dr/d\theta = 3 \cos \theta \). Setting this equal to zero gives \( \theta = \pi/2, 3\pi/2 \). These are the angles at which the tangents to the polar graph occur at the pole. In our case, this will be the x-axis and the negative x-axis, forming two tangents at the origin.

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Most popular questions from this chapter

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. \(r=\frac{6}{1+\cos \theta}\)

Show that the polar equation for \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta} \cdot \quad\) Ellipse

In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Parabola }} \quad \frac{\text { Eccentricity }}{e=1} \quad \frac{\text { Directrix }}{x=-1}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=3 \cos \theta, \quad y=3 \sin \theta $$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=1+\frac{1}{t}, \quad y=t-1 $$
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