2002 AIME II Problems/Problem 10
Contents
Problem
While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of for which the sine of
degrees is the same as the sine of
radians are
and
, where
,
,
, and
are positive integers. Find
.
Solution 1
Note that degrees is equal to
radians. Also, for
, the two least positive angles
such that
are
, and
.
Clearly for positive real values of
.
yields:
.
yields:
.
So, .
Solution 2
The first case is when the two angles, and
, are coterminal. The second case is when they are reflections of the
axis.
1.
for any integer
So
2. for any integer
So
Choosing carefully such that it's the minimum gives the answer same as above.
See also
2002 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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