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

Short Answer

Expert verified

The polar equation r=sinnθ is symmetric with respect to x-axis.

Hence it is proved.

Step by step solution

01

Given information

r=sinnθ

02

Calculation

Consider the polar equation where r=sinnθ,nis any integer.

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

Let (r,θ)be any point on the graph and f(θ)=sinnθwhere nis even.

By the definition of symmetry,

If a curve is symmetric with regard to the x-axis, then every point (r,θ)on the graph is symmetrical about the x-axis if(r,-θ) is also on the graph.

That means the point (r,θ)satisfies the relationship r=f(θ)then some point of the form (r,-θ+2nπ)or (-r,π-θ+2nπ)satisfies the relationship for some even integer n

That is r=f(-θ+2nπ) for some n then the function is symmetric about x-axis.

03

Calculation

Take the equation f(θ)=sinnθ

f(-θ)=sinn(-θ)f(-θ)=sin(-nθ)f(-θ)=sin(2π-nθ)

Then,

f(-θ)=sinnθ

As a result, every point on the graph(r,θ),(r,-θ) exists.

Therefore, the polar equation r=sinnθ is symmetric with respect to x-axis.

Hence it is proved.

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