Difference between revisions of "1991 AIME Problems/Problem 9"
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== Solution == | == Solution == | ||
− | Use the two [[Trigonometric identities#Pythagorean Identities|trigonometric Pythagorean identities]] <math> | + | == Solution 1 == |
+ | Use the two [[Trigonometric identities#Pythagorean Identities|trigonometric Pythagorean identities]] <math>1 + \tan^2 x = \sec^2 x</math> and <math>1 + \cot^2 x = \csc^2 x</math>. | ||
If we square <math>\sec x = \frac{22}{7} - \tan x</math>, we find that <math>\sec^2 x = \left(\frac{22}7\right)^2 - 2\left(\frac{22}7\right)\tan x + \tan^2 x</math>, so <math>1 = \left(\frac{22}7\right)^2 - \frac{44}7 \tan x</math>. Solving shows that <math>\tan x = \frac{435}{308}</math>. | If we square <math>\sec x = \frac{22}{7} - \tan x</math>, we find that <math>\sec^2 x = \left(\frac{22}7\right)^2 - 2\left(\frac{22}7\right)\tan x + \tan^2 x</math>, so <math>1 = \left(\frac{22}7\right)^2 - \frac{44}7 \tan x</math>. Solving shows that <math>\tan x = \frac{435}{308}</math>. | ||
− | Call <math>y = \frac mn</math>. Rewrite the second equation in a similar fashion: <math> | + | Call <math>y = \frac mn</math>. Rewrite the second equation in a similar fashion: <math>1 = y^2 - 2y\cot x</math>. Substitute in <math>\cot x = \frac{1}{\tan x} = \frac{308}{435}</math> to get a [[quadratic equation|quadratic]]: <math>0 = y^2 - \frac{616}{435} - 1</math>. The quadratic is [[factor]]able (though somewhat ugly); <math>(15y - 29)(29y + 15) = 0</math>. It turns out that only the [[positive]] root will work, so the value of <math>y = \frac{29}{15}</math> and <math>m + n = 044</math>. |
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+ | == Solution 2== | ||
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+ | This solution is fast for calculus students. Make the substitution <math>u = \tan \frac x2</math>. <math>\tan \frac x2 = \frac{\sin x}{1+\cos x}</math>, so <math>\csc x + \cot x = \frac{1+\cos x}{\sin x} = \frac1u = \frac mn</math>. <math>\sec x + \tan x = \frac{1 + \sin x}{\cos x}.</math> Now note the following: | ||
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+ | <math>\sin x = \frac{2u}{1+u^2}</math>; | ||
+ | <math>\cos x = \frac{1-u^2}{1+u^2}</math> | ||
+ | |||
+ | Plugging these into our equality gives: | ||
+ | |||
+ | <math>\frac{1+\frac{2u}{1+u^2}}{\frac{1-u^2}{1+u^2}} = \frac{22}7</math> | ||
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+ | This simplifies to <math>\frac{1+u}{1-u} = \frac{22}7</math>, and solving for <math>u</math> gives <math>u = \frac{15}{29}</math>, and <math>\frac mn = \frac{29}{15}</math>. Finally, <math>m+n = 044</math>. | ||
== See also == | == See also == |
Revision as of 11:47, 15 March 2008
Contents
[hide]Problem
Suppose that and that where is in lowest terms. Find
Solution
Solution 1
Use the two trigonometric Pythagorean identities and .
If we square , we find that , so . Solving shows that .
Call . Rewrite the second equation in a similar fashion: . Substitute in to get a quadratic: . The quadratic is factorable (though somewhat ugly); . It turns out that only the positive root will work, so the value of and .
Solution 2
This solution is fast for calculus students. Make the substitution . , so . Now note the following:
;
Plugging these into our equality gives:
This simplifies to , and solving for gives , and . Finally, .
See also
1991 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |