Difference between revisions of "1984 AIME Problems/Problem 13"
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<center><p><math>\tan((a+b)+(c+d)) = \frac{\frac{1}{2}+\frac{1}{8}}{1-\frac{1}{16}} = \frac{2}{3}</math>.</p></center> | <center><p><math>\tan((a+b)+(c+d)) = \frac{\frac{1}{2}+\frac{1}{8}}{1-\frac{1}{16}} = \frac{2}{3}</math>.</p></center> | ||
− | Thus our answer is <math>10\cdot\frac{3}{2}= | + | Thus our answer is <math>10\cdot\frac{3}{2}=\boxed{015}</math>. |
=== Solution 2 === | === Solution 2 === |
Revision as of 21:02, 15 December 2018
Problem
Find the value of
Contents
Solution
Solution 1
We know that so we can repeatedly apply the addition formula, . Let , , , and . We have
,
So
and
,
so
.
Thus our answer is .
Solution 2
Apply the formula repeatedly. Using it twice on the inside, the desired sum becomes . This sum can then be tackled by taking the cotangent of both sides of the inverse cotangent addition formula shown at the beginning.
Solution 3
On the coordinate plane, let , , , , , , , , , and . We see that , , , and . The sum of these four angles forms the angle of triangle , which has a cotangent of , which must mean that . So the answer is .
Solution 4
Recall that and that . Then letting and , we are left with
Expanding , we are left with
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |