Dear Uncle Colin,

I have to work out $\cot\left( \frac{3}{2}\pi \right)$. Wolfram Alpha says it’s 0, but when I work out $\frac{1}{\tan\left(\frac{3}{2}\pi\right)}$, my calculator shows an error. What’s going on?

- Troublesome Angle, No?

Hi, TAN, and thanks for your message!

The cotangent function is slightly unusual in that it appears to have two definitions: $\cot(x) = \frac{1}{\tan(x)}$ and $\cot(x) = \frac{\cos(x)}{\sin(x)}$.

Those look like they’re equivalent. And they are – everywhere except for odd multiples of $\piby 2$. There, you have a problem, because $\tan(x)$ is not defined there. A mathematical cowboy might say something like “well, $\tan\left(\piby 2\right)$ is infinity, and $\frac{1}{\infty} = 0$, so we’re all good,” but certainly not anywhere near the Mathematical Ninja.

Instead, it’s usually more sensible - if possibly a little more work - to say “I know $\cos\left( \frac{3}{2}\pi \right) = 0$, so $\cot\left( \frac{3}{2}\pi \right) = 0$.

Hope that helps!

- Uncle Colin