To graph the polar equation $r = 7\sin(\theta)$, here's an explanation of how it behaves:

1. Equation Type: This is a polar equation of a limaçon. Since the coefficient of $\sin(\theta)$ is positive, it will be symmetric about the vertical axis.

2. Behavior:

   • When $\theta = 0$, $r = 7\sin(0) = 0$.
   • When $\theta = \pi/2$, $r = 7\sin(\pi/2) = 7$.
   • When $\theta = \pi$, $r = 7\sin(\pi) = 0$.
   • When $\theta = 3\pi/2$, $r = 7\sin(3\pi/2) = -7$. Negative $r$ indicates the graph points to the opposite direction.

3. Graph Description:

   • The curve forms a circle since the equation $r = 7\sin(\theta)$ is a special case of a limaçon that simplifies to a circle.

Let me create a graph for better visualization.import numpy as np import matplotlib.pyplot as plt

Define the polar equation

theta = np.linspace(0, 2 * np.pi, 500) r = 7 * np.sin(theta)

Create the polar plot

plt.figure(figsize=(8, 8)) ax = plt.subplot(111, polar=True) ax.plot(theta, r, label=r'$r = 7\sin(\theta)$')

Enhance the plot

ax.set_title("Polar Graph of r = 7sin(θ)", va='bottom', fontsize=14) ax.legend(loc='upper right') plt.show() <<ImageDisplayed>>successHere is the graph of the polar equation $r = 7\sin(\theta)$. The curve forms a circle centered at $(0, 3.5)$ in the polar coordinate system, as expected from the equation.

Would you like further details about this graph or its properties?

Here are five follow-up questions you might consider:

1. What are the properties of polar equations involving $\sin$ or $\cos$?
2. How does the graph change if we multiply $\sin(\theta)$ by a constant?
3. How would the graph change if we replaced $\sin(\theta)$ with $\cos(\theta)$?
4. What if the constant $7$ is replaced with another value, such as a negative number?
5. How can we convert this polar equation to Cartesian coordinates?

Tip: Polar graphs are often symmetric about the origin, horizontal axis, or vertical axis, depending on whether they involve $\sin$ or $\cos$.