In the unit circle, find the coordinates of the point corresponding to an angle of $frac{5pi}{6}$ radians. Explain your steps and reasoning.
Answer 1
Emily Hall
To find the coordinates of the point corresponding to an angle of $\frac{5\pi}{6}$ radians on the unit circle, we need to find the cosine and sine of the angle.
First, observe that $\frac{5\pi}{6}$ radians is in the second quadrant.
The reference angle is $\pi – \frac{5\pi}{6} = \frac{\pi}{6}$.
In the second quadrant, the cosine is negative and the sine is positive.
Thus, the coordinates are $(-\cos \frac{\pi}{6}, \sin \frac{\pi}{6})$.
$\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$ and $\sin \frac{\pi}{6} = \frac{1}{2}$.
Therefore, the coordinates are $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.
Answer 2
William King
To determine the coordinates for the angle $frac{5pi}{6}$ on the unit circle, we calculate the corresponding trigonometric functions:
Recognize that $frac{5pi}{6}$ places the angle in the second quadrant.
Calculate the reference angle: $pi – frac{5pi}{6} = frac{pi}{6}$.
In the second quadrant, cosine is negative, and sine is positive. Therefore:
$cos left(frac{5pi}{6}
ight) = -cos left(frac{pi}{6}
ight)$ and $sin left(frac{5pi}{6}
ight) = sin left(frac{pi}{6}
ight)$.
Given $cos left(frac{pi}{6}
ight) = frac{sqrt{3}}{2}$ and $sin left(frac{pi}{6}
ight) = frac{1}{2}$,
the coordinates are: $left(-frac{sqrt{3}}{2}, frac{1}{2}
ight)$.
Answer 3
Henry Green
To find the coordinates for angle $frac{5pi}{6}$ on the unit circle, use the fact that:
$ ext{Reference angle} = pi – frac{5pi}{6} = frac{pi}{6}$.
In the second quadrant, $cos$ is negative, $sin$ is positive:
$cos left(frac{pi}{6}
ight) = frac{sqrt{3}}{2}$, so:
$cos left(frac{5pi}{6}
ight) = -frac{sqrt{3}}{2}$.
$sin left(frac{pi}{6}
ight) = frac{1}{2}$, so:
$sin left(frac{5pi}{6}
ight) = frac{1}{2}$.
Coordinates: $left(-frac{sqrt{3}}{2}, frac{1}{2}
ight)$.
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