5. Each of the following equations is given in terms of polar coordinates. Rewrite eachequation using rectangular coordinates only (that is, without using the symbols r and θ ). Also, graph each equation on one of the polar coordinate grids given below.(a) r=5 (b) rcosθ = 3(c) θ = 1/3 π **Transcription of Educational Content: Polar Coordinates**### Problem 5Each of the following equations is given in terms of polar coordinates. Rewrite each equation using rectangular coordinates only (that is, without using the symbols \( r \) and \( \theta \)). Also, graph each equation on one of the polar coordinate grids given below.(a) \( r = 5 \)(b) \( r \cos \theta = 3 \)(c) \( \theta = \frac{1}{3} \pi \)### Polar Grids ExplainedThe diagrams consist of polar coordinate grids, which are circular and divided into 360 degrees. Each grid has concentric circles representing different radii and radial lines representing angles in degrees. The horizontal line marks \( \theta = 0^\circ \), which aligns with the positive x-axis in rectangular coordinates.### Detailed Description- **Graph (a):** Represents a circle with a constant radius of 5 centered at the origin. - **Graph (b):** Represents a vertical line at \( x = 3 \), illustrating the conversion \( r \cos \theta = x \).- **Graph (c):** Depicts a line where the angle \( \theta \) is constant, \( \theta = \frac{1}{3} \pi \), which corresponds to a line radiating out from the origin at approximately 60 degrees.### Graphing Polar CoordinatesStudying polar coordinates often involves learning about various curves like the polar rose or limacon. In this activity, we focus on visual understanding and conversion of these equations into rectangular forms. Understanding these principles helps clarify why certain curves appear as they do and how to graph them effectively. This topic will also cover memorizing foundational curves like the cardioid and other functions within this coordinate system.

(a) r=5 We know x=rcosθy=rsinθ∴x2+y2=r2sin2θ+cos2θ⇒x2+y2=r2 ∵sin2θ+cos2θ=1⇒x2+y2=52⇒x2+y2=25 Hence the equation in rectangular co-ordinate system is x2+y2=25 Therefore the equation is x2+y2=25(b) rcosθ=3 We know that x=rcosθ⇒x=3 ∵rcosθ=3 Hence the equation in rectangular co-ordinate system is x=3(c) θ=π3 x=rcosθy=rsinθ Given θ=π3 x=rcosπ3⇒x=r2y=rsinπ3⇒y=r32⇒2y=2x3 ∵x=r2⇒y=3x Hence the equation in rectangular co-ordinate system is y=x3

Math

Advanced Math

165° 180° 195* 210° 150° 135 .522 5. Each of the following equations is given in terms of polar coordinates. Rewrite each equation using rectangular coordinates only (that is, without using the symbols r and ). Also, graph each equation on one of the polar coordinate grids given below. 120° .002 (a) (b) (c) 105° 592 ģ -OLZ in 582 60° .00€ r=5 rcose = 3 1 0=-n 37 45° 30° 15" 345 330° 315* 0=0° 180° 165 195* 210⁰ 150° 135 .522 120° 105 2012 .592 -OLZ 582 60° 45° +00E STE 30° 15° 345 330* 8=0°

165° 180° 195* 210° 150° 135 .522 5. Each of the following equations is given in terms of polar coordinates. Rewrite each equation using rectangular coordinates only (that is, without using the symbols r and ). Also, graph each equation on one of the polar coordinate grids given below. 120° .002 (a) (b) (c) 105° 592 ģ -OLZ in 582 60° .00€ r=5 rcose = 3 1 0=-n 37 45° 30° 15" 345 330° 315* 0=0° 180° 165 195* 210⁰ 150° 135 .522 120° 105 2012 .592 -OLZ 582 60° 45° +00E STE 30° 15° 345 330* 8=0°

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

5. Each of the following equations is given in terms of polar coordinates. Rewrite each equation using rectangular coordinates only (that is, without using the symbols r and θ ). Also, graph each equation on one of the polar coordinate grids given below. (a) r=5 (b) rcosθ = 3 (c) θ = 1/3 π

**Transcription of Educational Content: Polar Coordinates**

### Problem 5
Each of the following equations is given in terms of polar coordinates. Rewrite each equation using rectangular coordinates only (that is, without using the symbols \( r \) and \( \theta \)). Also, graph each equation on one of the polar coordinate grids given below.

(a) \( r = 5 \)

(b) \( r \cos \theta = 3 \)

(c) \( \theta = \frac{1}{3} \pi \)

### Polar Grids Explained

The diagrams consist of polar coordinate grids, which are circular and divided into 360 degrees. Each grid has concentric circles representing different radii and radial lines representing angles in degrees. The horizontal line marks \( \theta = 0^\circ \), which aligns with the positive x-axis in rectangular coordinates.

### Detailed Description

- **Graph (a):** Represents a circle with a constant radius of 5 centered at the origin.

- **Graph (b):** Represents a vertical line at \( x = 3 \), illustrating the conversion \( r \cos \theta = x \).

- **Graph (c):** Depicts a line where the angle \( \theta \) is constant, \( \theta = \frac{1}{3} \pi \), which corresponds to a line radiating out from the origin at approximately 60 degrees.

### Graphing Polar Coordinates

Studying polar coordinates often involves learning about various curves like the polar rose or limacon. In this activity, we focus on visual understanding and conversion of these equations into rectangular forms. Understanding these principles helps clarify why certain curves appear as they do and how to graph them effectively.

This topic will also cover memorizing foundational curves like the cardioid and other functions within this coordinate system.

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