I am supposed to identify the classical curve that is represented by

r = 1 + sin Theta.

I think it's a cardioid, because its equation is r = 1 + 1 sin theta (r = a + a sin theta). Is this correct?

that is correct

wolframalpha . com will draw it for you. Just type in

plot r = 1 + sin(theta)

thank you so much for your time.

Yes, you are correct! The curve represented by the equation r = 1 + sin Theta is indeed a cardioid. Let me explain how to arrive at this conclusion.

To identify the classical curve represented by a polar equation, it can be helpful to simplify the equation and analyze its form. In this case, the equation r = 1 + sin Theta can be rewritten as r = a + a sin Theta, where a = 1.

By comparing this form with the general equation for a cardioid, r = a + a sin Theta, we can see that the given equation matches the form of a cardioid. The value of a determines the size of the cardioid, but the shape remains the same.

A cardioid is a curve that resembles a stylized heart shape. It is defined by the equation r = a + a sin Theta, where a is the distance from the tip of the cardioid to the center (usually called the cusp). In this case, since a = 1, the equation represents a cardioid with a distance of 1 unit from the cusp to the center.

So, in conclusion, the equation r = 1 + sin Theta represents a cardioid.