The following problem appeared on the American High School Mathematics Examination (now called the AMC 12) in 1988:
If , what is ?
When I presented this problem to a group of students, I was pleasantly surprised by the amount of creativity shown when solving this problem.
Yesterday, I presented a solution using a Pythagorean identity, but I was unable to be certain if the final answer was a positive or negative without drawing a picture. Here’s a third solution that also use a Pythagorean trig identity but avoids this difficulty. Again, I begin by squaring both sides.
Yesterday, I used the Pythagorean identity again to find . Today, I’ll instead plug back into the original equation :
Unlike the example yesterday, the signs of and must agree. That is, if , then must also be positive. On the other hand, if , then must also be negative.
If they’re both positive, then
and if they’re both negative, then
Either way, the answer must be .
This is definitely superior to the solution provided in yesterday’s post, as there’s absolutely no doubt that the product must be positive.