/*! 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 134 Convert the polar equation \(r=\... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 134

Convert the polar equation \(r=\cos \theta+3 \sin \theta\) to rectangular form and identify the graph.

Short Answer

Expert verified
The rectangular form of the given polar equation is \((x - \frac{1}{2})^2 + (y - \frac{3}{2})^2 = \frac{5}{4}\). The graph represents a circle with a center at (0.5, 1.5) and a radius of \(\frac{\sqrt{5}}{2}\)

Step by step solution

01

Express r in terms of x and y

Start by expressing r in terms of x and y using the relationship \(r = \sqrt{x^2 + y^2}\) to get \(\sqrt{x^2 + y^2} = \cos\theta + 3\sin\theta\)
02

Subtitute \(\theta\)

Now replace \(\cos\theta\) and \(\sin\theta\) with \(x/r\) and \(y/r\) respectively to get \(\sqrt{x^2 + y^2} = \frac{x}{\sqrt{x^2 + y^2}} + \frac{3y}{\sqrt{x^2 + y^2}}\)
03

Simplify the equation

Simplify the equation by multiplying every term by \(\sqrt{x^2 + y^2}\) to eliminate the denominator to get \(x^2 + y^2 = x + 3y\)
04

Manipulate the equation

Rearrange this equation into the standard form of an equation which might help identify its graph easier to get \(x^2 - x + y^2 - 3y = 0\) which can be rewritten as \((x - \frac{1}{2})^2 + (y - \frac{3}{2})^2 = \frac{5}{4}\)
05

Identify the graph

This equation is in the standard form for a circle. Therefore, the graph is a circle with center at \((0.5, 1.5)\) and a radius of \(\frac{\sqrt{5}}{2}\)

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Most popular questions from this chapter

In Exercises \(91-116\), convert the polar equation to rectangular form. $$\theta=2 \pi / 3$$

A projectile is launched at a height of \(h\) feet above the ground at an angle of \(\theta\) with the horizontal. The initial velocity is \(v_{0}\) feet per second, and the path of the projectile is modeled by the parametric equations $$x=\left(v_{0} \cos \theta\right) t$$ and $$y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}.$$ Use a graphing utility to graph the paths of a projectile launched from ground level at each value of \(\boldsymbol{\theta}\) and \(v_{0} .\) For each case, use the graph to approximate the maximum height and the range of the projectile. (a) \(\theta=15^{\circ}, \quad v_{0}=50\) feet per second (b) \(\theta=15^{\circ}, \quad v_{0}=120\) feet per second (c) \(\theta=10^{\circ}, \quad v_{0}=50\) feet per second (d) \(\theta=10^{\circ}, \quad v_{0}=120\) feet per second

Convert the polar equation $$r=2(h \cos \theta+k \sin \theta)$$ to rectangular form and verify that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle.

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=\frac{5}{1-4 \cos \theta}$$

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=\frac{6}{2 \cos \theta-3 \sin \theta}$$
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