Find the values of $cot heta$ on the unit circle

The cotangent function is defined as the ratio of the cosine of an angle to the sine of the angle: $\cot \theta = \frac{\cos \theta}{\sin \theta}$

Given that the angle $\theta$ is located at $\frac{\pi}{4}$ radians on the unit circle, we need to find the value of $\cot \frac{\pi}{4}$.

Using the unit circle values:

$\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$

Therefore,

$\cot \frac{\pi}{4} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$

To find $cot heta$ at $ heta = frac{5pi}{6}$, we use the definition $cot heta = frac{cos heta}{sin heta}$

From the unit circle:

$cos frac{5pi}{6} = -frac{sqrt{3}}{2}, sin frac{5pi}{6} = frac{1}{2}$

Thus,

$cot frac{5pi}{6} = frac{-frac{sqrt{3}}{2}}{frac{1}{2}} = -sqrt{3}$

To determine $cot heta$ when $ heta = pi$, we use:

$cot heta = frac{cos heta}{sin heta}$

From the unit circle:

$cos pi = -1, sin pi = 0$

Since the sine of ( pi ) is zero, $cot pi$ is undefined.

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