Chapter 9: Q 60. (page 748)

Prove that, for every integer $n$ the graph of $r = \cos n θ$ is symmetrical with respect to the $x -$ axis.

Short Answer

Expert verified

The polar equation $r = \cos n θ$ is symmetric with respect to $x -$ axis.

Step by step solution

Given information

$r = \cos n θ$

Calculation

Consider the polar equation $r = \cos n θ$

The goal is to show that the equation is symmetric around the $x -$ axis.

If for any point $(r, θ)$ on the graph, the point $(r, - θ)$ is also on the graph, the curve is symmetric with respect to the $x -$ axis.

By the definition of symmetry,

A curve is symmetric with respect to the $x -$ axis if, for every point $(r, θ)$ on the graph, the point $(r, - θ)$ is also on the graph.

Calculation

Take the equation $f (θ) = \cos n θ$ where $n$ any integer

$f (- θ) = \cos n (- θ) = \cos n θ$

As a result, the point $(r, θ)$ is also on the graph, as is the point $(r, - θ)$

As a result, with regard to the $x -$ axis, the polar equation $r = \cos n θ$ is symmetric.

Hence it is proved.

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Prove that, for every integer $n$ the graph of $r = \cos n θ$ is symmetrical with respect to the $x -$ axis.

Short Answer

Step by step solution

Given information

Calculation

Calculation

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Prove that, for every integer n the graph of r=cosnθ is symmetrical with respect to the x-axis.

Short Answer

Step by step solution

Given information

Calculation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Prove that, for every integer $n$ the graph of $r = \cos n θ$ is symmetrical with respect to the $x -$ axis.