Determine the exact values of $cot( heta)$ for an angle $ heta$ that, when doubled, corresponds to a point on the unit circle where the y-coordinate is equal to $-frac{sqrt{3}}{2}$.
Answer 1
To find $\cot(\theta)$, we need to determine the appropriate angle $2\theta$. Given that the y-coordinate of $2\theta$ is $-\frac{\sqrt{3}}{2}$, we know that $2\theta$ corresponds to $240^\circ$ or $300^\circ$ in the unit circle.
1. For $2\theta = 240^\circ$:
$\theta = \frac{240^\circ}{2} = 120^\circ$
$\cot(120^\circ) = \cot(180^\circ – 60^\circ) = -\cot(60^\circ) = -\frac{1}{\sqrt{3}}$
2. For $2\theta = 300^\circ$:
$\theta = \frac{300^\circ}{2} = 150^\circ$
$\cot(150^\circ) = \cot(180^\circ – 30^\circ) = -\cot(30^\circ) = -\sqrt{3}$
Therefore, the exact values of $\cot(\theta)$ are $-\frac{1}{\sqrt{3}}$ and $-\sqrt{3}$.
Answer 2
To solve for $cot( heta)$, we need to find the angle $2 heta$ such that the y-coordinate is $-frac{sqrt{3}}{2}$. This occurs at $2 heta = 240^circ$ or $2 heta = 300^circ$.
For $2 heta = 240^circ$:
$ heta = frac{240^circ}{2} = 120^circ$
$cot(120^circ) = -cot(60^circ) = -frac{1}{sqrt{3}}$
For $2 heta = 300^circ$:
$ heta = frac{300^circ}{2} = 150^circ$
$cot(150^circ) = -cot(30^circ) = -sqrt{3}$
Hence, $cot( heta)$ is $-frac{1}{sqrt{3}}$ or $-sqrt{3}$.
Answer 3
To find $cot( heta)$, we need $2 heta$ such that the y-coordinate is $-frac{sqrt{3}}{2}$. This occurs at $240^circ$ or $300^circ$.
1. $2 heta = 240^circ$:
$ heta = 120^circ$
$cot(120^circ) = -frac{1}{sqrt{3}}$
2. $2 heta = 300^circ$:
$ heta = 150^circ$
$cot(150^circ) = -sqrt{3}$
Therefore, $cot( heta)$ is $-frac{1}{sqrt{3}}$ or $-sqrt{3}$.
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