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.

Short answer

Answer: For odd integer values of \(m\), the graph represents a rose with \(m\) petals for both polar equations. For even integer values of \(m\), the graph represents a rose with \(2m\) petals for both polar equations.

Step-by-step solution

Analyze the polar equation in terms of sine function

For \(r = a \sin{m\theta}\), we will analyze two cases: odd integer \(m\) and even integer \(m\). We know that the sine function oscillates between \(-1\) and \(1\). Therefore, when \(m\theta\) varies from \(0\) to \(2\pi\), we can expect some kind of symmetrical shape to form.

Graph the polar equation \(r = a \sin{m\theta}\) for odd integer \(m\)

For odd integer values of \(m\), let's first consider \(m = 1\). In this case, we have \(r = a \sin{\theta}\). As \(\theta\) varies from \(0\) to \(2\pi\), the graph of this polar equation will represent a single loop. If we choose \(m = 3\), for example, we get the polar equation \(r = a \sin{3\theta}\). As \(\theta\) varies from \(0\) to \(2\pi\), we will get three loops in the graph, forming a 3-petal rose. Similarly, for any odd integer value of \(m\), the graph will represent a rose with \(m\) petals.

Graph the polar equation \(r = a \sin{m\theta}\) for even integer \(m\)

For even integer values of \(m\), let's first consider \(m = 2\). In this case, we have the polar equation \(r = a \sin{2\theta}\). As \(\theta\) varies from \(0\) to \(2\pi\), the graph of this polar equation will represent a double loop (i.e., 4 petals). If we choose \(m = 4\), the graph will display a rose with 8 petals. So, for even integer values of \(m\), the graph will represent a rose with \(2m\) petals.

Analyze the polar equation in terms of cosine function

Now let's analyze the polar equation \(r = a \cos{m\theta}\). As with the sine function, the cosine function oscillates between \(-1\) and \(1\). Thus, we can expect similar patterns and symmetries as for the sine function.

Graph the polar equation \(r = a \cos{m\theta}\) for odd integer \(m\)

For odd integer values of \(m\), let's first choose \(m = 1\). The polar equation \(r = a \cos{\theta}\) will represent a single loop, just like with the sine function. If we choose \(m = 3\), the polar equation \(r = a \cos{3\theta}\) will describe a 3-petal rose. Similarly, for any odd integer value of \(m\), the graph will represent a rose with \(m\) petals.

Graph the polar equation \(r = a \cos{m\theta}\) for even integer \(m\)

For even integer values of \(m\), let's first choose \(m = 2\). The polar equation \(r = a \cos{2\theta}\) will represent a rose with 4 petals. If we choose \(m = 4\), the graph will represent a rose with 8 petals. So, for even integer values of \(m\), the graph will represent a rose with \(2m\) petals.

Conclusion

For both polar equations, \(r = a \sin{m\theta}\) and \(r = a \cos{m\theta}\), we have shown that for odd integer values of \(m\), the graph represents a rose with \(m\) petals, whereas for even integer values of \(m\), the graph represents a rose with \(2m\) petals.