91Ó°ÊÓ

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.