Question

Graph the following equations. Use a graphing utility to check your work and produce a final graph. $$r=\sin ^{2}(\theta / 2)$$

Short answer

#Answer# To graph the polar equation $$r=\sin^{2}(\theta / 2)$$, first, identify key points, specifically when r = 0 and r = 1. For r = 0, θ is any multiple of π; and for r = 1, θ is any odd multiple of π. Then, sketch a graph by plotting r = 0 at every multiple of π and r = 1 at every odd multiple of π. This will create alternating loops with different orientations. To verify the result, use a graphing utility like Desmos or Mathematica to compare the hand-drawn graph with the generated graph.

Step-by-step solution

Review polar coordinates basics

Polar coordinates are a method of representing points in the plane in terms of a radial distance, r, and an angle, θ. It is essential to understand how to convert between polar coordinates to Cartesian coordinates and vice versa. We will use this knowledge to identify some key points on the graph.

Identify key features or points

By analyzing the polar equation $$r = \sin^{2}({\theta\over 2})$$, plot some key points to help in sketching the graph. Using the trigonometric identity $$0 \leq \sin ^{2}(x) \leq 1$$, the radial distance r varies between 0 and 1. Note down these key points: - When $$\sin^{2}(\frac{\theta}{2}) = 0$$, then r = 0. - When $$\sin^{2}(\frac{\theta}{2}) = 1$$, then r = 1. Now, solve for θ-values for these conditions: 1. When r = 0, $$\sin^{2}(\frac{\theta}{2}) = 0$$. This means $$\sin(\frac{\theta}{2}) = 0$$, so $$\frac{\theta}{2} = k\pi$$ where k is an integer. And θ = 2kπ, or any multiple of π. 2. When r = 1, $$\sin^{2}(\frac{\theta}{2}) = 1$$. This means $$\sin(\frac{\theta}{2}) = \pm 1$$, so $$\frac{\theta}{2} = \frac{(2k+1)}{2}\pi$$ where k is an integer. And θ = (2k + 1)π, or any odd multiple of π. With these key points, we have enough information to sketch the polar graph.

Sketch the polar graph

Begin by plotting the key points identified in step 2 on the polar plane. Remember that r represents the radial distance from the origin, and θ represents the angle counterclockwise from the positive x-axis. 1. Plot r = 0 at every multiple of π. 2. Plot r = 1 at every odd multiple of π. Because sine is a periodic function and we have covered the r ranges between 0 and 1 and θ for multiples of π, the graph will create alternating loops with different orientations. Start by plotting the first few loops and continue the pattern. Now, you have a hand-drawn graph for the polar function $$r = \sin^{2}(\frac{\theta}{2})$$.

Verify with a graphing utility

Use a graphing utility like Desmos or Mathematica to plot the polar equation $$r = \sin^{2}(\frac{\theta}{2})$$, and compare the result with your hand-drawn graph. Your graph should match the shape and pattern of the graph generated by the utility.