# How do you find the value of cot 90?

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Erudite Share
Jul 11, 2015

$\cot \left(90\right) = 0$

#### Explanation:

Recall that $\cot \left(\theta\right) = \frac{1}{\tan} \left(\theta\right)$

and that $\tan \left(\theta\right) = \sin \frac{\theta}{\cos} \left(\theta\right)$

So,

$\cot \left(\theta\right) = \frac{1}{\tan} \left(\theta\right) = \frac{1}{\sin \frac{\theta}{\cos} \left(\theta\right)} = \cos \frac{\theta}{\sin} \left(\theta\right)$

Now, let's just put in 90 degrees for $\theta$

$\cot \left(\theta\right) = \cos \frac{\theta}{\sin} \left(\theta\right)$

$\cot \left(90\right) = \cos \frac{90}{\sin} \left(90\right)$

Recall, from the unit circle (below) that $\sin \left(90\right) = 1$ and $\cos \left(90\right) = 0$:

So,

$\cot \left(90\right) = \cos \frac{90}{\sin} \left(90\right)$

$\cot \left(90\right) = \frac{0}{1} = 0$

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