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Chapter 8: Problem 31

Convert the polar equation to rectangular form and sketch its graph. $$ r=\theta $$

Short Answer

Expert verified
The rectangular equation, in the form of parametric equations, are x = 胃 cos(胃) and y = 胃 sin(胃). The graph of the equation starts from the origin and spirals outwards, with the distance from the origin (r) always equal to the angle (胃).

Step by step solution

01

Converting the equation from polar to rectangular form

Use the relationships x = r cos(胃) and y = r sin(胃) to convert the polar equation r = 胃 to rectangular form. According to the polar equation, the radius or distance from the origin (r) is equal to the angle (胃). Substitute r = 胃 into both equations to convert the form: \[ x = 胃 \cos(胃) \] \[ y = 胃 \sin(胃) \]
02

Recognize the parametric form

The equations obtained in Step 1 form a parametric equation in 胃. where x and y are represented as functions of 胃. It has no simple equation in x and y, but some of its properties can be deduced from the polar equation.
03

Sketching the graph

To graph the equation in rectangular coordinates, note that for every value of the angle 胃, the distance r from the origin is equal to the angle. The graph therefore will start from the origin and spiral outwards, because as 胃 increases, so does the radius.

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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. $$ x=\sqrt{t}, \quad y=t-2 $$

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \text { Hyperbola: } x=h+a \sec \theta, \quad y=k+b \tan \theta $$

Give the integral formulas for area and arc length in polar coordinates.

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

Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? (a) \(x=t+1, y=t^{3}\) (b) \(x=-t+1, y=(-t)^{3}\)
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