Monday, April 9, 2007

Get A Tan. Topic: Trigonometry. Level: AMC.

Problem: (2007 MAO State - Gemini) Let $ x $ be in degrees and $ 0^{\circ} < x < 45^{\circ} $. Solve for $ x $: $ \tan{(4x)} = \frac{\cos{x}-\sin{x}}{\cos{x}+\sin{x}} $.

Solution: Here's a nice tangent identity that is not very well-known, but rather cool. Start with the regular tangent angle addition identity,

$ \tan{(x+y)} = \frac{\tan{x}+\tan{y}}{1-\tan{x} \cdot \tan{y}} $.

Letting $ x = 45^{\circ} $ and $ y = -\theta $, we obtain

$ \tan{(45^{\circ}-\theta)} = \frac{1-\tan{\theta}}{1+\tan{\theta}} = \frac{\cos{\theta}-\sin{\theta}}{\cos{\theta}+\sin{\theta}} $.

Well, look at that. It's the same expression as the RHS of the equation we want to solve. Substituting accordingly, it remains to solve

$ \tan{(4x)} = \tan{(45^{\circ}-x)} \Rightarrow 4x = 45^{\circ}-x \Rightarrow x = 9^{\circ} $.



Comment: This identity is a nice one to keep around because it can turn up unexpectedly. Especially when you see that exact form and you're like "whoa this is such a nice form there must be an identity for it." So there.


Practice Problem: (2007 MAO State - Gemini) One hundred positive integers, not necessarily distinct, have a sum of $ 331 $. What is the largest possible product these numbers can attain?


  1. The general problem is not hard. Ideally we want to partition a sum into e's. We can't do that, so some mix of 2's and 3's produces maximality. Blah blah blah logs blah blah blah.

  2. That works when you don't have a limit on the number of positive integers. Here, a mix of 2's and 3's will definitely not sum to 331 ;).