Find the exact values of $cot$ for specific angles on the unit circle

To find the exact values of $\cot$ for specific angles on the unit circle, let’s consider the angle $\theta = \frac{11\pi}{6}$.

Step 1: Identify the coordinates on the unit circle: The angle $\theta = \frac{11\pi}{6}$ corresponds to the point $\left( \cos(\frac{11\pi}{6}), \sin(\frac{11\pi}{6}) \right)$.

Step 2: Use the coordinates to find the cotangent: We know $\cos(\frac{11\pi}{6}) = \frac{\sqrt{3}}{2}$ and $\sin(\frac{11\pi}{6}) = -\frac{1}{2}$.

Step 3: Calculate $\cot(\theta)$: Using $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$, we get

$ \cot\left( \frac{11\pi}{6} \right) = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} $

Let’s find the value of $cot$ for the angle $ heta = frac{7pi}{4}$ on the unit circle.

Step 1: Determine the coordinates on the unit circle: The angle $ heta = frac{7pi}{4}$ corresponds to the point $left( cos(frac{7pi}{4}), sin(frac{7pi}{4})
ight)$.

Step 2: Use the coordinates to find the cotangent: We know $cos(frac{7pi}{4}) = frac{sqrt{2}}{2}$ and $sin(frac{7pi}{4}) = -frac{sqrt{2}}{2}$.

Step 3: Calculate $cot( heta)$: Using $cot( heta) = frac{cos( heta)}{sin( heta)}$, we get

$ cotleft( frac{7pi}{4}
ight) = frac{frac{sqrt{2}}{2}}{-frac{sqrt{2}}{2}} = -1 $

To find the value of $cot$ for the angle $ heta = frac{5pi}{3}$:

Step 1: Identify the coordinates: The angle $ heta = frac{5pi}{3}$ corresponds to $left( cos(frac{5pi}{3}), sin(frac{5pi}{3})
ight)$.

Step 2: Coordinates are $cos(frac{5pi}{3}) = frac{1}{2}$ and $sin(frac{5pi}{3}) = -frac{sqrt{3}}{2}$.

Step 3: Calculate $cot( heta)$:

$ cotleft( frac{5pi}{3}
ight) = frac{frac{1}{2}}{-frac{sqrt{3}}{2}} = -frac{1}{sqrt{3}} = -frac{sqrt{3}}{3} $

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