Problem 23 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r=2-3 \sin \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 2 - 3\sin(\theta)\) is symmetrical with respect to the y-axis, but not with respect to the x-axis. The graph has a maximum radius of 5 and a minimum radius of -1.

Step 1: Test for Symmetry About the X-axis

Replace \(\theta\) with \(-\theta\) in the equation and simplify: \(r = 2 - 3\sin(-\theta)\). Due to the odd property of the sine function, \(\sin(-\theta) = -\sin(\theta)\). Therefore, the equation becomes: \(r = 2 + 3\sin(\theta)\). This is not identical to the original equation, \(r = 2 - 3\sin(\theta)\), so the graph is not symmetrical about the x-axis.

Step 2: Test for Symmetry About the Y-axis

Replace \(\theta\) with \(\pi - \theta\) in the equation and simplify: \(r = 2 - 3\sin(\pi - \theta)\). Due to the properties of the sine function in the second quadrant, \(\sin(\pi - \theta) = \sin(\theta)\). Therefore, the equation remains the same: \(r = 2 - 3\sin(\theta)\). This is identical to the original equation, so the graph is indeed symmetrical about the y-axis.

Step 3: Graph the Polar Equation

With polar graphs, begin by determining key values such as the maximum, minimum, and zeros of the function. From \(r = 2 - 3\sin(\theta)\), observe that the maximum value for the radius occurs when \(\sin(\theta) = -1\), giving \(r = 2 - 3(-1) = 5\). The minimum occurs when \(\sin(\theta) = 1\), giving \(r = 2 - 3(1) = -1\). Use these values to help draw a plot. Create a symmetrical plot about the y-axis based on findings from Step 2. Remember that 'r' can be negative in polar coordinates, which means the point lies in the opposite direction of the angle.