Kary created the table below to graph the equation r = 1 + 2 sin θ (rounded to the hundredths place). Kary thinks she made a mistake.
This equation represents a polar equation in the form of r = a + b sin θ, where a is the radius when θ = 0 and b determines the amplitude of the sine curve.
When b is positive, the graph will oscillate between the values a - |b| and a + |b|. In this case, a = 1 and b = 2, so the radius oscillates between -1 and 3.
Here is the corrected table for the equation r = 1 + 2 sin θ:
| θ | r |
|------|----------------|
| 0 | 1 + 2(0) = 1 |
| π/4 | 1 + 2(√2/2) = 1 + √2 ≈ 2.41 |
| π/2 | 1 + 2(1) = 3 |
| 3π/4 | 1 + 2(√2/2) = 1 + √2 ≈ 2.41 |
| π | 1 + 2(0) = 1 |
| 5π/4 | 1 + 2(-√2/2) = 1 - √2 ≈ -0.41 |
| 3π/2 | 1 + 2(-1) = -1 |
| 7π/4 | 1 + 2(-√2/2) = 1 - √2 ≈ -0.41 |
| 2π | 1 + 2(0) = 1 |
By plotting these points correctly, you will see that the graph oscillates between -0.41 and 2.41, with the peak values occurring at θ = π/2 and 3π/2.