Identify and graph the rose given by the polar equation \(r = 5\cos(3\theta)\).
Answer
Step 3: Sketch the rose. Start at the origin and sketch a petal extending 5 units in the positive x-direction. Then sketch the other two petals, each rotated by \(120\) degrees from the previous one. The resulting graph will be a rose with 3 petals.
Step 1 ：Step 1: Identify the properties of the rose. The number of petals of a polar rose is determined by the coefficient of \(\theta\) in the equation. If the coefficient is even, the rose will have twice as many petals. If it is odd, the number of petals will be the same as the coefficient. In this case, the coefficient of \(\theta\) is 3, which is odd, so the rose will have 3 petals.
Step 2 ：Step 2: Determine the length of the petals. The length of the petals is determined by the coefficient of the trigonometric function. In this case, the coefficient is 5, so each petal will extend 5 units from the origin.
Step 3 ：Step 3: Sketch the rose. Start at the origin and sketch a petal extending 5 units in the positive x-direction. Then sketch the other two petals, each rotated by \(120\) degrees from the previous one. The resulting graph will be a rose with 3 petals.
