Problem 99 - Calculus Early Transcendentals

Question

Show that the graph of \(r=a \sin m \theta\) or \(r=a \cos m \theta\) is a rose with \(m\) leaves if \(m\) is an odd integer and a rose with \(2 m\) leaves if \(m\) is an even integer.

Step-by-Step Solution

Answer

Answer: For the given polar equations, the rose graph will have \(m\) leaves if \(m\) is an odd integer and \(2m\) leaves if \(m\) is an even integer.

Step 1: Basic understanding and properties

Let's first understand the given polar equations, \(r = a\sin {m\theta}\) and \(r = a\cos {m\theta}\). Here, \(r\) is the distance from the origin (pole) to a point on the graph, \(a\) is the amplitude, and \(m\) is the number of leaves. \(\theta\) is the polar angle.

Step 2: Analyze the equation when \(m\) is an odd integer

Let's analyze the case where \(m\) is odd. We know that the sine and cosine functions have a range from -1 to 1. Therefore, for odd integers, the distance from the origin \(r\) would oscillate between \(-a\) and \(a\) as \(\theta\) varies. In one revolution \((0\leq \theta \leq 2\pi)\), \(\sin{m\theta}\) and \(\cos {m\theta}\) reach their maximum value \(m\) times. Hence the equation will reach \(r=a\) or \(r=-a\) exactly \(m\) times, resulting in \(m\) leaves for the rose graph.

Step 3: Analyze the equation when \(m\) is an even integer

Now, let's analyze the case where \(m\) is even. In this case, sine and cosine functions will only output positive values when multiplied by an even number, causing r to oscillate between 0 and \(a\) as \(\theta\) varies. Similar to the odd integer case, we find that the equation reaches its maximum value \(m\) times in one revolution \((0\leq \theta \leq 2\pi)\). However, since the sine and cosine functions only output positive values, each leaf is "doubled", creating a total of \(2m\) leaves for the rose graph.

Step 4: Conclusion

In conclusion, we have shown that for a given polar equation of type \(r = a\sin {m\theta}\) or \(r = a\cos {m\theta}\), we can obtain a rose graph with \(m\) leaves if \(m\) is an odd integer and a rose with \(2m\) leaves if \(m\) is an even integer.