3

I have an equation $$x = \csc(\theta) - \cot(\theta).$$

As $\theta$ approaches zero, $x$ approaches zero. However, trying to solve the equation at zero yields an undefined result.

How do I rewrite the equation to be continuous at 0?

Arturo Magidin
  • 398,050
MerickOWA
  • 133

1 Answers1

5

Hint: Write the cosecant and cotangent in terms of sine and cosine. You can then combine the two fractions to give an expression that goes to 0/0. Expanding in a Taylor series or L'Hopital's rule will then be your friend.

Ross Millikan
  • 374,822
  • 4
    Instead of Taylor series or l'Hôpital's rule, one can use the fact that $1 - \cos\theta = 2\sin^2(\theta/2)$, and then expand $\sin\theta$ in terms of $\theta/2$ as well. –  Jan 18 '11 at 16:42
  • Indeed, you can only treat this properly in terms of limits. – Noldorin Jan 18 '11 at 16:43
  • 1
    @Rahul Narain ah! thats it. So rewriting and canceling gives me $\tan(\theta/2)$! – MerickOWA Jan 18 '11 at 16:59