Final answer:
To sketch the polar equation r = 3 - 3sinθ, calculate r for various values of θ and plot them on a polar coordinate system. The graph will resemble a limaçon due to the sinusoidal relationship between r and θ.
Step-by-step explanation:
The student has asked to sketch the graph of the polar equation r = 3 - 3sinθ. In polar coordinates, the equation can be used to find the distance r from the origin to a point and the angle θ that this line makes with the x-axis. To sketch the graph, you would typically plot several points by choosing values for θ (which can range from 0 to 2π) and calculating the corresponding r values, then mark these points on a polar coordinate system. Since r is dependent on the sine of θ, the graph will have a sinusoidal shape, and in this case, it describes a type of limaçon.
To convert from polar to Cartesian coordinates (x, y), you would use the relationships x = r × cos(θ) and y = r × sin(θ). This is because the r sin θ factor represents the y-component (vertical) of the polar vector, and the r cos θ factor represents the x-component (horizontal).