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Chapter 9: Q 60. (page 748)

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

Short Answer

Expert verified

The polar equation r=cosn胃 is symmetric with respect tox-axis.

Step by step solution

01

Given information

r=cosn胃

02

Calculation

Consider the polar equation r=cosn胃

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

If for any point (r,胃) on the graph, the point (r,-胃) 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 thex-axis if, for every point (r,胃) on the graph, the point (r,-胃) is also on the graph.

03

Calculation

Take the equation f(胃)=cosn胃where nany integer

f(-胃)=cosn(-胃)=cosn胃

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

As a result, with regard to the x-axis, the polar equation r=cosn胃 is symmetric.

Hence it is proved.

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