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Prove that the area enclosed by one petal of the polar rose r=cos3θis the same as the area enclosed by one petal of the polar rose r=sin3θ.

Short Answer

Expert verified

It is proved that the area enclosed by one petal of the polar rose r = cos 3θ is the same as the area enclosed by one petal of the polar rose r = sin 3θ.

Step by step solution

01

Step 1. Given information

Two petals of the polar rose are r = cos 3θ and r = sin 3θ.

02

Step 2. Explanation

Consider the petal r=cos3θ

Area localid="1652009009589" =2120π6cos3θ2dθ

localid="1652009016020" =120π61+cos6θdθ=12π6+sin6π66-0=π12

Consider the petal width="67">r=sin3θ

Area =120π3sin3θ2dθ

=120π31-cos6θ2dθ=14θ-sin6θ60π3=π12

So, the area enclosed by one petal of the polar rose r = cos 3θ is the same as the area enclosed by one petal of the polar rose r = sin 3θ.

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