This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

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\sin(3\theta)$.
You're asking for a visual representation of the graphs. Since I can't draw images directly, I'll provide a detailed description of how to sketch each graph based on the calculations we just did.
a) Sketch the graph of .
To sketch the graph of :
Step 1: Set up the coordinate axes. Draw a horizontal axis for and a vertical axis for . Label the -axis with increments of (e.g., ). Label the -axis with values from -1 to 1 (e.g., -1, 0, 1).
Step 2: Plot the key points for one cycle (). • Start at the origin: Plot a point at . • Rise to the maximum: Plot a point at . • Return to the axis: Plot a point at . • Fall to the minimum: Plot a point at . • Return to the axis: Plot a point at .
Step 3: Draw the curve. Connect these plotted points with a smooth, wave-like curve. The curve should start at , go up to , come down through , continue down to , and then rise back up to . This completes one full cycle of the sine wave.
b) Sketch the graph of .
To sketch the graph of :
Step 1: Set up the coordinate axes. Draw a horizontal axis for and a vertical axis for . Label the -axis with increments of (e.g., ). Label the -axis with values from -1 to 1 (e.g., -1, 0, 1).
Step 2: Plot the key points for one cycle (). • Start at the maximum: Plot a point at . • Cross the axis: Plot a point at . • Fall to the minimum: Plot a point at . • Cross the axis: Plot a point at . • Return to the maximum: Plot a point at .
Step 3: Draw the curve. Connect these plotted points with a smooth, wave-like curve. The curve should start at , go down through , continue down to , then rise through , and finally go up to . This completes one full cycle of the cosine wave.
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You're asking for a visual representation of the graphs. Since I can't draw images directly, I'll provide a detailed description of how to sketch each graph based on the calculations we just did.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.