Problem 37 Test for symmetry and then graph... [FREE SOLUTION]

Chapter 7: Problem 37

Test for symmetry and then graph each polar equation. $$r=\sin \theta+\cos \theta$$

Short Answer

Expert verified

The polar equation $r= \sin \theta + \cos \theta$ is neither symmetric about the polar axis, nor the origin, nor the line $ \theta=\pi/2 $. Its graph is a straight line passing through the pole and making an angle $ \theta=\frac{\pi}{4} $ with the polar axis.

Step by step solution

Test Symmetry about the Polar Axis

This is done by replacing $ \theta $ with $ -\theta $ in the equation. If the equation remains unchanged, it is symmetric about the polar axis. So, \Substitute $ \theta = -\theta $ into the original equation to get $ r = \sin(-\theta) + \cos(-\theta) $. Using the properties of sine and cosine for negative angles, we rewrite this as $ r = -\sin(\theta) - \cos(\theta) $. Since this differs from the original equation, the graph of this equation is not symmetric about the polar axis.

Test Symmetry about the Origin

This is done by replacing $ r $ with $ -r $ in the equation. If the equation remains unchanged, it is symmetric about the origin. But here, replacing $ r $ with $ -r $ does not produce the original equation, so it shows that the graph of the equation is not symmetric about the origin.

Test Symmetry about the Line $ \theta=\pi/2 $

This is done by replacing $ \theta $ with $ \pi - \theta $ in the equation. If the equation remains unchanged, it is symmetric about the line $ \theta=\pi/2 $. But upon substitution, the equation is not unaffected, therefore, it's not symmetric under this condition

Convert Polar Equation to Rectangular Coordinates

Remember that in polar coordinates, $ x=r\cos \theta $ and $ y = r\sin \theta $. So the equation $ r=\sin \theta + \cos \theta $ can be rewritten as $ r = \frac{y}{r} + \frac{x}{r} $ which simplifies to $ x + y = r $. At this stage, the equation is in a simple and understandable format that can be plotted easily.

Graph the Equation

The rectangular form of the equation can be graphed using basic knowledge of linear equations. We know this is a straight line with an x-intercept and a y-intercept at r. Identify at least two points and draw the line. You will notice that it's a straight line passing through the pole (origin) making an angle $ \theta=\frac{\pi}{4} $ with the polar axis.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Test for symmetry and then graph each polar equation. $$r=\sin \theta+\cos \theta$$

Short Answer

Step by step solution

Test Symmetry about the Polar Axis

Test Symmetry about the Origin

Test Symmetry about the Line \( \theta=\pi/2 \)

Convert Polar Equation to Rectangular Coordinates

Graph the Equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Probability and Statistics

Statistics

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.