1986 AIME Problems/Problem 3

Revision as of 13:38, 6 May 2007 by I_like_pie (talk | contribs) (See also)

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 $\tan$:

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

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