Question

Graph the following equations. Use a graphing utility to check your work and produce a final graph. $$r^{2}=16 \sin 2 \theta$$

Short answer

Question: Graph the polar equation r²=16sin(2θ) using appropriate steps. Answer: To graph the polar equation r²=16sin(2θ), follow these steps: 1. Identify the type of polar curve (symmetry): The graph will be symmetrical about the polar axis (the θ = 0 line) due to the presence of the sine function. 2. Make a table of values for r and θ: Create a table with values for θ ranging from 0 to π and calculate the corresponding r² and r. | θ | 2θ | sin(2θ) | r² | r | |---------|---------|---------|---------|------| | 0 | 0 | 0 | 0 | 0 | | π/6 | π/3 | √3/2 | 8√3 | 2√6 | | π/3 | 2π/3 | 1 | 16 | 4 | | π/2 | π | 0 | 0 | 0 | | 2π/3 | 4π/3 | -1 | 16 | -4 | | 5π/6 | 5π/3 | -√3/2 | 8√3 | -2√6 | | π | 2π | 0 | 0 | 0 | 3. Plot the points and sketch the curve: Plot the points using the table and connect them with smooth curves to complete the graph. 4. Verify the result using a graphing utility: Input the polar equation r²=16sin(2θ) into a graphing utility such as Desmos or a TI-graphing calculator to ensure that the graph produced matches the one created in Step 3.

Step-by-step solution

Identify the type of polar curve (symmetry)

Since our polar equation contains a sine function sin(2θ), we can expect that the graph will be symmetrical about the polar axis (the θ = 0 line).

Make a table of values for r and θ

First, let's create a table of values for θ and r. Since the sine function has a period of π, we can expect our polar equation to have values between 0 and π. Create a table with values for θ ranging from 0 to π, incrementing in the most appropriate steps (in this case, let's use π/6), and calculate the corresponding r² and r: | θ | 2θ | sin(2θ) | r² | r | |---------|--------|---------|-----|------| | 0 | 0 | 0 | 0 | 0 | | π/6 | π/3 | √3/2 | 8√3 | 2√6 | | π/3 | 2π/3 | 1 | 16 | 4 | | π/2 | π | 0 | 0 | 0 | | 2π/3 | 4π/3 | -1 | 16 | -4 | | 5π/6 | 5π/3 | -√3/2 | 8√3 | -2√6 | | π | 2π | 0 | 0 | 0 |

Plot the points and sketch the curve

Now, we want to plot our r, θ values in polar coordinates. Remember that to plot polar coordinates (r, θ), we move r units away from the origin in the direction θ. Let's plot the points using the table we've created: - (0, 0) - (2√6, π/6) - (4, π/3) - (0, π/2) - (-4, 2π/3) - (-2√6, 5π/6) - (0, π) After plotting the points, connect them with smooth curves to complete the graph.

Verify the result using a graphing utility

Finally, to verify that the graph is correct, we can use a graphing utility such as Desmos or a TI-graphing calculator. Input the polar equation r²=16sin(2θ) and ensure that the graph produced by the utility matches the one created in Step 3.