객관식
If the initial side of an angle in standard position lies along the positive -axis and is rotated clockwise to coincide with the negative -axis, by what angle (in ) has it turned?
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1. Measuring Angles
Angles in Standard Position
Multiple Choice
Given the polar coordinate , which of the following points in rectangular coordinates represents the same location?
A
B
C
D
검증된 단계별 안내1
Recall that polar coordinates are given as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
To convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), use the formulas:
\(x = r \cos(\theta)\)
\(y = r \sin(\theta)\)
In this problem, the polar coordinate is given as \((-6, \frac{\pi}{12})\). Notice that the radius \(r\) is negative, which means the point lies in the direction opposite to the angle \(\theta\).
A negative radius \(r\) can be handled by adding \(\pi\) to the angle \(\theta\) and using the positive radius \(|r|\). So, rewrite the point as \((6, \frac{\pi}{12} + \pi)\).
Now apply the conversion formulas with \(r = 6\) and \(\theta = \frac{\pi}{12} + \pi\):
\(x = 6 \cos\left(\frac{\pi}{12} + \pi\right)\)
\(y = 6 \sin\left(\frac{\pi}{12} + \pi\right)\)
This gives the rectangular coordinates representing the same location.
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