Question

A student observes that the polar equation r = 2 - 3sin(0) produces a
maçon with an inner loop. Which statement best explains why an inner loop forms and what it represents?
* 1 point
An inner loop forms because sin(e) oscillates between -1 and 1, creating multiple intersections with the polar axis
An inner loop forms because the coefficient of sin(0) exceeds the constant term in absolute value, causing r to become negative for certain angle values, which reflects points across the origin
An inner loop forms because the equation is undefined for certain values of 0 between 0 and n
An inner loop forms because the constant term is positive while the sine coefficient is negative, creating a symmetric pattern

Answer 1

Option 2.

Because |−3| > |2|, r = 2 − 3 sinθ can be negative for some θ (r ranges from 2−3 = −1 to 2+3 = 5). Negative r means the point is placed in the opposite direction (reflected across the origin), so the curve doubles back and creates an inner loop. The inner loop is exactly the portion of the graph traced when r < 0.