Find the value of $cot heta$ for a given angle on the unit circle

Given an angle $\theta = \frac{5\pi}{6}$, we need to find the value of $\cot \theta$.

The coordinates of the point corresponding to $\theta = \frac{5\pi}{6}$ on the unit circle are $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$.

From the definition of cotangent, $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

Here, $\cos \theta = -\frac{\sqrt{3}}{2}$ and $\sin \theta = \frac{1}{2}$.

Therefore,

$\cot \theta = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}}$

$ = -\sqrt{3}$

So, $\cot \left(\frac{5\pi}{6}\right) = -\sqrt{3}$.

Given an angle $ heta = frac{7pi}{4}$, we need to find the value of $cot heta$.

The coordinates of the point corresponding to $ heta = frac{7pi}{4}$ on the unit circle are $(frac{sqrt{2}}{2}, -frac{sqrt{2}}{2})$.

From the definition of cotangent, $cot heta = frac{cos heta}{sin heta}$.

Here, $cos heta = frac{sqrt{2}}{2}$ and $sin heta = -frac{sqrt{2}}{2}$.

Therefore,

$cot heta = frac{frac{sqrt{2}}{2}}{-frac{sqrt{2}}{2}}$

$ = -1$

So, $cot left(frac{7pi}{4}
ight) = -1$.

Given an angle $ heta = frac{pi}{3}$, we need to find the value of $cot heta$.

The coordinates of the point corresponding to $ heta = frac{pi}{3}$ on the unit circle are $(frac{1}{2}, frac{sqrt{3}}{2})$.

From the definition of cotangent, $cot heta = frac{cos heta}{sin heta}$.

Here, $cos heta = frac{1}{2}$ and $sin heta = frac{sqrt{3}}{2}$.

Therefore,

$cot heta = frac{frac{1}{2}}{frac{sqrt{3}}{2}}$

$ = frac{1}{sqrt{3}}$

So, $cot left(frac{pi}{3}
ight) = frac{1}{sqrt{3}}$.

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