Problem

Identify the type of rose and graph it for the following polar equation: r=3cos⁡(2θ)

Answer

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Answer

To graph this rose, we plot points for various angles θ from 0 to 2π and connect these points smoothly. The points will trace out two loops, each of radius 3, with one loop for 0≤θ≤π and the other for π≤θ≤2π.

Steps

Step 1 ：The rose is specified by the polar equation r=acos⁡(nθ) or r=asin⁡(nθ), where a is the length of the petal and n is the number of petals if n is even or twice the number of petals if n is odd.

Step 2 ：From the given polar equation, r=3cos⁡(2θ), we can identify a=3 and n=2. So, this is a rose with 2 petals and each petal has a length of 3.

Step 3 ：To graph this rose, we plot points for various angles θ from 0 to 2π and connect these points smoothly. The points will trace out two loops, each of radius 3, with one loop for 0≤θ≤π and the other for π≤θ≤2π.

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