Difference between revisions of "1986 AIME Problems/Problem 3"
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Since <math>\cot</math> is the reciprocal function of <math>\tan</math>: | Since <math>\cot</math> is the reciprocal function of <math>\tan</math>: | ||
− | <math> | + | <math>\cot x + \cot y = \frac{1}{\tan x} + \frac{1}{\tan y} = \frac{\tan x + \tan y}{\tan x \cdot \tan y} = 30</math> |
Thus, <math>\tan x \cdot \tan y = \frac{\tan x + \tan y}{30} = \frac{25}{30} = \frac{5}{6}</math> | Thus, <math>\tan x \cdot \tan y = \frac{\tan x + \tan y}{30} = \frac{25}{30} = \frac{5}{6}</math> | ||
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<math>\tan(x+y) = \frac{\tan x + \tan y}{1-\tan x \cdot \tan y} = \frac{25}{1-\frac{5}{6}} = 150</math> | <math>\tan(x+y) = \frac{\tan x + \tan y}{1-\tan x \cdot \tan y} = \frac{25}{1-\frac{5}{6}} = 150</math> | ||
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== See also == | == See also == | ||
{{AIME box|year=1986|num-b=2|num-a=4}} | {{AIME box|year=1986|num-b=2|num-a=4}} |
Revision as of 17:19, 13 February 2008
Problem
If and , what is ?
Solution
Since is the reciprocal function of :
Thus,
Using the tangent addition formula:
See also
1986 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |