1988 AHSME Problems/Problem 13
Problem
If then what is ?
Solution
In the problem we are given that , and we want to find . We can divide both sides of the original equation by to get We can now use right triangle trigonometry to finish the problem.
(Note that this assumes that is acute. If is obtuse, then is positive and is negative, so the equation cannot be satisfied. If is reflex, then both and are negative, so the equation is satisfied, but when we find , the two negatives will cancel out and give the same (positive) answer as in the acute case.)
Since the problem asks us to find . So is our answer.
Solution 2 (Pure Algebra)
Squaring both sides gives . We can take the Pythagorean identity, and substitute the 1st equation in, giving . So , and .
Multiplying the 2 together gives , and then taking the square root gives . However, implies one of and is negative while the other is positive, but means they have the same sign, which contradicts the first statement. This means is the only answer.
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See also
1988 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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