Ask Uncle Colin: A Calculator Error

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 $\frac{\pi }{2}$. There, you have a problem, because $\tan (x)$ is not defined there. A mathematical cowboy might say something like “well, $\tan \left(\frac{\pi }{2}\right)$ is infinity, and $\frac{1}{\mathrm{\infty }}=0$, 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