Multiple Choice
On the unit circle, what is the radian measure of the arc that subtends a central angle of ?
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3. Unit Circle
Trigonometric Functions on the Unit Circle
Multiple Choice
Given the polar equation , which of the following best describes the shape of its graph on the unit circle?
A
A circle centered at the origin
B
A straight line through the origin
C
A lemniscate (figure-eight shape)
D
A four-petaled rose curve
Verified step by step guidance1
Start by examining the given polar equation: \(r^{2} = \sin(2\theta)\). Notice that \(r^{2}\) is expressed in terms of a trigonometric function of \(2\theta\), which suggests symmetry and periodicity in the graph.
Recall that equations of the form \(r^{2} = \sin(n\theta)\) or \(r^{2} = \cos(n\theta)\) often represent rose curves, where the number of petals depends on the value of \(n\). Specifically, if \(n\) is even, the rose has \$2n\( petals; if \)n\( is odd, it has \)n$ petals.
In this problem, \(n = 2\), so the rose curve will have \(2 \times 2 = 4\) petals. This matches the description of a four-petaled rose curve.
Understand that the shape is not a circle or a straight line because those would have simpler forms such as \(r = \text{constant}\) for a circle or \(\theta = \text{constant}\) for a line. The presence of \(\sin(2\theta)\) and the squared radius indicates a more complex, petal-like shape.
Therefore, by analyzing the form of the equation and the value of \(n\), you conclude that the graph represents a four-petaled rose curve.
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