# Mathematical Food for Thought

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Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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Don’t You Wish You Remembered Those Trig Identities. Topic: Trigonometry. Level: AMC/AIME. February 9th, 2007

Problem: (1962 IMO – #4) Solve the equation .

Solution: Subtract the from both sides, and replace the LHS by , so we now have

.

Recalling the sine angle addition identity, we write

.

So our equation is

.

Expanding, collecting terms, and simplifying using and , we get

.

Divide by , rearrange, and factor:

.

Substitute using the double-angle identities and to finally find

.

Solving yields

for all integers . QED.

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Comment: Pretty standard and slightly ugly manipulation of trig identities, but overall not too bad. Considering you get like an hour or more to do this, I don’t feel too bad about it. Other solutions taking into account symmetry or using DeMoivre’s and stuff are also out there, but this seemed much more straightforward and easier to develop.

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Practice Problem: (2007 AMC 12A – #24) For each integer , let be the number of solutions of the equation on the interval . What is ?

### 3 Responses to “Don’t You Wish You Remembered Those Trig Identities. Topic: Trigonometry. Level: AMC/AIME.”

1. blue_giraffe Says:

i remembered them but there’s still no way i could have gotten that =)

2. paladin8 Says:

Maybe after a lot of substituting and messing around? =P

3. silverwolf Says:

If only problems on the IMO today could be like this… not saying I’m getting into the IMO… just saying

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