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

Short Answer

Expert verified

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

Step by step solution

01

Given information

r=sinnθ

02

Calculation

Consider the polar coordinate system function r=sinnθ

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 θπ-θ or θ-θ

03

Calculation

Consider the transformation θπ-θ in the function r=sinnθ

r=sinn(π-θ)=sin(nπ-nθ)

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

That is, r=sinnθ

Thus, the function does not change under the transformation θπ-θ

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

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