1991 AIME Problems/Problem 9
Problem
Suppose that and that where is in lowest terms. Find
Contents
[hide]Solution
Solution 1
Use the two trigonometric Pythagorean identities and .
If we square , we find that , so . Solving shows that .
Call . Rewrite the second equation in a similar fashion: . Substitute in to get a quadratic: . This factors as . It turns out that only the positive root will work, so the value of and .
Solution 2
Make the substitution (a substitution commonly used in calculus). , so . Now note the following: Plugging these into our equality gives:
This simplifies to , and solving for gives , and . Finally, .
See also
1991 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |