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>\displaystyle \cot x + \cot y = \frac{1}{\tan x} + \frac{1}{\tan x} = \frac{\tan x + \tan y}{\tan x \cdot \tan y} = 30</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 $\displaystyle \tan x+\tan y=25$ and $\displaystyle \cot x + \cot y=30$, what is $\displaystyle \tan(x+y)$?

Solution

Since $\cot$ is the reciprocal function of $\tan$:

$\cot x + \cot y = \frac{1}{\tan x} + \frac{1}{\tan y} = \frac{\tan x + \tan y}{\tan x \cdot \tan y} = 30$

Thus, $\tan x \cdot \tan y = \frac{\tan x + \tan y}{30} = \frac{25}{30} = \frac{5}{6}$

Using the tangent addition formula:

$\tan(x+y) = \frac{\tan x + \tan y}{1-\tan x \cdot \tan y} = \frac{25}{1-\frac{5}{6}} = 150$

See also

1986 AIME (ProblemsAnswer KeyResources)
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