Difference between revisions of "1986 AIME Problems/Problem 3"
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== Problem == | == Problem == | ||
+ | If <math>\tan x+\tan y=25</math> and <math>\cot x + \cot y=30</math>, what is <math>\tan(x+y)</math>? | ||
− | == Solution == | + | == Solution 1 == |
+ | Since <math>\cot</math> is the reciprocal function of <math>\tan</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> | ||
+ | |||
+ | Using the tangent addition formula: | ||
+ | |||
+ | <math>\tan(x+y) = \frac{\tan x + \tan y}{1-\tan x \cdot \tan y} = \frac{25}{1-\frac{5}{6}} = \boxed{150}</math>. | ||
+ | |||
+ | == Solution 2 == | ||
+ | |||
+ | Using the formula for tangent of a sum, <math>\tan(x+y)=\frac{\tan x + \tan y}{1-\tan x \tan y} = \frac{25}{1-\tan x \tan y}</math>. We only need to find <math>\tan x \tan y</math>. | ||
+ | |||
+ | We know that <math>25 = \tan x + \tan y = \frac{\sin x}{\cos x} + \frac{\sin y}{\cos y}</math>. Cross multiplying, we have <math>\frac{\sin x \cos y + \cos x \sin y}{\cos x \cos y} = \frac{\sin(x+y)}{\cos x \cos y} = 25</math>. | ||
+ | |||
+ | Similarly, we have <math>30 = \cot x + \cot y = \frac{\cos x}{\sin x} + \frac{\cos y}{\sin y} = \frac{\cos x \sin y + \sin x \cos y}{\sin x \sin y} = \frac{\sin(x+y)}{\sin x \sin y}</math>. | ||
+ | |||
+ | Dividing: | ||
+ | |||
+ | <math>\frac{25}{30} = \frac{\frac{\sin(x+y)}{\cos x \cos y}}{\frac{\sin(x+y)}{\sin x \sin y}} = \frac{\sin x \sin y}{\cos x \cos y} = \tan x \tan y = \frac{5}{6}</math>. Plugging in to the earlier formula, we have <math>\tan(x+y) = \frac{25}{1-\frac{5}{6}} = \frac{25}{\frac{1}{6}} = \boxed{150}</math>. | ||
+ | |||
+ | == Solution 3 (less trig required, use of quadratic formula) == | ||
+ | |||
+ | Let <math>a=\tan x</math> and <math>b=\tan y</math>. This simplifies the equations to: | ||
+ | |||
+ | <cmath>a + b = 25</cmath> | ||
+ | |||
+ | <cmath>\frac{1}{a} + \frac{1}{b} = 30</cmath> | ||
+ | |||
+ | Taking the tangent of a sum formula from Solution 2, we get <math>\tan(x+y) = \frac{25}{1 - ab}</math>. | ||
+ | |||
+ | We can use substitution to solve the system of equations from above: <math>b = -a + 25</math>, so <math>\frac{1}{a} + \frac{1}{-a + 25} = 30</math>. | ||
+ | |||
+ | Multiplying by <math>-a(a-25)</math>, we get <math>a + (-a + 25) = -30a(a-25)</math>, which is <math>-30a^2 + 750a = 25</math>. Dividing everything by 5 and shifting everything to one side gives <math>6a^2 - 150a + 5 = 0</math>. | ||
+ | |||
+ | Using the quadratic formula gives <math>a = \frac{150 \pm \sqrt {22380}}{12}</math>. Since this looks too hard to simplify, we can solve for <math>b</math> using <math>a + b = 25</math>, which turns out to also be <math>b = \frac{150 \pm \sqrt {22380}}{12}</math>, provided that the sign of the radical in <math>a</math> is opposite the one in <math>b</math>. | ||
+ | |||
+ | WLOG, assume <math>a = \frac{150 + \sqrt{22380}}{12}</math> and <math>b = \frac{150 - \sqrt{22380}}{12}</math>. Multiplying them gives <math>ab = \frac{22500 - 22380}{144}</math> which simplifies to <math>\frac{5}{6}</math>. | ||
+ | |||
+ | THe denominator of <math>\frac{25}{1 - ab}</math> ends up being <math>\frac{1}{6}</math>, so multiplying both numerator and denominator by 6 gives <math>\boxed{150}</math>. | ||
+ | |||
+ | -ThisUsernameIsTaken | ||
+ | |||
+ | |||
+ | |||
+ | Remark: The quadratic need not be solved. The value of <math>ab</math> can be found through Vieta's. | ||
== See also == | == See also == | ||
− | |||
− | |||
{{AIME box|year=1986|num-b=2|num-a=4}} | {{AIME box|year=1986|num-b=2|num-a=4}} | ||
+ | * [[AIME Problems and Solutions]] | ||
+ | * [[American Invitational Mathematics Examination]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 22:44, 12 October 2023
Contents
Problem
If and , what is ?
Solution 1
Since is the reciprocal function of :
Thus,
Using the tangent addition formula:
.
Solution 2
Using the formula for tangent of a sum, . We only need to find .
We know that . Cross multiplying, we have .
Similarly, we have .
Dividing:
. Plugging in to the earlier formula, we have .
Solution 3 (less trig required, use of quadratic formula)
Let and . This simplifies the equations to:
Taking the tangent of a sum formula from Solution 2, we get .
We can use substitution to solve the system of equations from above: , so .
Multiplying by , we get , which is . Dividing everything by 5 and shifting everything to one side gives .
Using the quadratic formula gives . Since this looks too hard to simplify, we can solve for using , which turns out to also be , provided that the sign of the radical in is opposite the one in .
WLOG, assume and . Multiplying them gives which simplifies to .
THe denominator of ends up being , so multiplying both numerator and denominator by 6 gives .
-ThisUsernameIsTaken
Remark: The quadratic need not be solved. The value of can be found through Vieta's.
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 |
- AIME Problems and Solutions
- American Invitational Mathematics Examination
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.