Find bounds on θ that trace out the specified piece of function only once.
Since the function uses sin(3θ), it's a reasonable guess that values of θ in the family will be useful.
See what we find for .
That's a full petal, but not the one we want.
That's the part we want. The bounds for θ are .
Start with 0 ≤ θ ≤ π, since that seems to work a lot:
That's not enough graph. Try expanding the bounds for θ.
How about 0 ≤ θ ≤ 2π?
That's the right piece of the graph. The bounds we want are 0 ≤ θ ≤ 2π.
Try 0 ≤ θ ≤ π:
No, that's too much graph. Since there's a in the argument to cosine.
Try knocking the upper bound down to .
That's not enough graph.
Try doubling the upper bound of θ, we have :
That's what we want. The final answer is .
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