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Chapter 9: Q 58. (page 748)

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

Short Answer

Expert verified

The function r=sinn胃 is symmetric about the y-axis for all values of n

Step by step solution

01

Given information

r=sinn胃

02

Calculation

Consider the polar coordinate system function r=sinn胃

Here, nis any integer.

The objective is to prove that the graph of the function is symmetric about the y-axis for every integer n

The graph in a polar coordinate system is symmetric about the y-axis if it is invariant with respect to the transformation 胃鈫�蟺-胃 or 胃鈫�-胃

03

Calculation

Consider the transformation 胃鈫�蟺-胃 in the function r=sinn胃

r=sinn(蟺-胃)=sin(n蟺-n胃)

For any integer n the angle (n蟺-n胃) lies in the second quadrant and in the second quadrant sin function is symmetric.

That is, r=sinn胃

Thus, the function does not change under the transformation 胃鈫�蟺-胃

Hence, the function r=sinn胃 is symmetric about the y-axis for all values of n

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