Given the point $P$ on the unit circle at an angle of $210^{circ}$, find $cos(210^{circ})$ and $sin(210^{circ})$.
Answer 1
Chloe Evans
To find $\cos(210^{\circ})$ and $\sin(210^{\circ})$, we start by converting the angle to radians:
$210^{\circ} = 210 \cdot \frac{\pi}{180} = \frac{7\pi}{6}$
The reference angle for $\frac{7\pi}{6}$ is $30^{\circ}$ or $\frac{\pi}{6}$.
The coordinates of a point on the unit circle at an angle $\theta$ in the third quadrant are $(-\cos(\theta), -\sin(\theta))$.
Since $\cos(30^{\circ}) = \frac{\sqrt{3}}{2}$ and $\sin(30^{\circ}) = \frac{1}{2}$, we get:
$\cos(210^{\circ}) = -\cos(30^{\circ}) = -\frac{\sqrt{3}}{2}$
$\sin(210^{\circ}) = -\sin(30^{\circ}) = -\frac{1}{2}$
Therefore, the coordinates of the point are:
$P(\cos(210^{\circ}), \sin(210^{\circ})) = \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$
Answer 2
Charlotte Davis
First, convert $210^{circ}$ to radians:
$210^{circ} = 210 cdot frac{pi}{180} = frac{7pi}{6}$
The reference angle for $frac{7pi}{6}$ is $30^{circ}$.
Since $210^{circ}$ is in the third quadrant, both $cos$ and $sin$ will be negative.
$cos(30^{circ}) = frac{sqrt{3}}{2}$
$sin(30^{circ}) = frac{1}{2}$
Thus,
$cos(210^{circ}) = -frac{sqrt{3}}{2}$
$sin(210^{circ}) = -frac{1}{2}$
Answer 3
Lucas Brown
Convert $210^{circ}$ to radians:
$210^{circ} = frac{7pi}{6}$
Since $cos(30^{circ}) = frac{sqrt{3}}{2}$ and $sin(30^{circ}) = frac{1}{2}$, and $210^{circ}$ is in the third quadrant, we have:
$cos(210^{circ}) = -frac{sqrt{3}}{2}$
$sin(210^{circ}) = -frac{1}{2}$
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