Difference between revisions of "1996 AIME Problems/Problem 10"
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Since <math>19^2 \equiv 361 \equiv 1 \pmod{180}</math>, multiplying both sides by <math>19</math> yields <math>x \equiv 141 \cdot 19 \equiv (140+1)(18+1) \equiv 0+140+18+1 \equiv 159 \pmod{180}</math>. | Since <math>19^2 \equiv 361 \equiv 1 \pmod{180}</math>, multiplying both sides by <math>19</math> yields <math>x \equiv 141 \cdot 19 \equiv (140+1)(18+1) \equiv 0+140+18+1 \equiv 159 \pmod{180}</math>. | ||
− | Therefore, the smallest positive solution is <math>x = \boxed{159}</math>. | + | Therefore, the smallest positive solution is <math>x = \boxed{159}</math>. |
+ | == Solution 2 == | ||
+ | <math>\dfrac{\cos{96^\circ}}+\sin{96^{\circ}}}{\cos96^{\circ}}-\sin{96^{\circ}}}=<cmath>\dfrac{1+\tan{96^{\circ}}}{1-\tan{96^{\circ}}} = </cmath>\drac{\tan{45^{\circ}}+\tan{96^{\circ}}}{1-\tan{45^{\circ}}\tan{96^{\circ}}} = </math><math> \tan{141^{\circ}}</math> | ||
+ | |||
+ | So <math>19x = 141 +180n</math>, for some integer <math>n</math>. | ||
+ | Then <math>0 \equiv 8 + 9n \pmod{19}</math>, from which <math>n \equiv 16 \pmod{19}</math> after some manipulation. The smallest suitable value of <math>n</math> is therefore 16 from which <math>x = \boxed{159}</math> | ||
== See also == | == See also == |
Revision as of 20:00, 17 August 2015
Contents
Problem
Find the smallest positive integer solution to .
Solution
.
The period of the tangent function is , and the tangent function is one-to-one over each period of its domain.
Thus, .
Since , multiplying both sides by
yields
.
Therefore, the smallest positive solution is .
Solution 2
$\dfrac{\cos{96^\circ}}+\sin{96^{\circ}}}{\cos96^{\circ}}-\sin{96^{\circ}}}=<cmath>\dfrac{1+\tan{96^{\circ}}}{1-\tan{96^{\circ}}} = </cmath>\drac{\tan{45^{\circ}}+\tan{96^{\circ}}}{1-\tan{45^{\circ}}\tan{96^{\circ}}} =$ (Error compiling LaTeX. Unknown error_msg)
So , for some integer
.
Then
, from which
after some manipulation. The smallest suitable value of
is therefore 16 from which
See also
1996 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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