1991 AIME Problems/Problem 9
Problem
Suppose that and that
where
is in lowest terms. Find
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 |