Difference between revisions of "1989 AIME Problems/Problem 10"
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+ | === Solution 3=== | ||
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+ | Use Law of cosines to give us <math>c^2=a^2+b^2-2ab\cos(\gamma)</math> or therefore <math>\cos(\gamma)=\frac{994c^2}{ab}</math>. Next, we are going to put all the sin's in term of <math>\sin(a)</math>. We get <math>\sin(\gamma)=\frac{c\sin(a)}{a}</math>. Therefore, we get <math>\cot(\gamma)=\frac{994c}{b\sina}</math>. | ||
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+ | Next, use Law of Cosines to give us <math>b^2=a^2+c^2-2ac\cos(\beta)</math>. Therefore, <math>\cos(\beta)=\frac{a^2-994c^2}{ac}</math>. Also, <math>\sin(\beta)=\frac{b\sin(a)}{a}</math>. Hence, <math>\cot(\beta)=\frac{a^2-994c^2}{bc\sin(a)}</math>. | ||
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+ | Lastly, <math>\cos(\alpha)=\frac{b^2-994c^2}{bc}</math>. Therefore, we get <math>\cot(\alpha)=\frac{b^2-994c^2}{bc\sin(a)}</math>. | ||
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+ | Now, <math>\frac{\cot(\gamma)}{\cot(\beta)+\cot(\alpha)}=\frac{\frac{994c}{b\sina}}{\frac{a^2-994c^2+b^2-994c^2}{bc\sin(a)}}</math>. After using <math>a^2+b^2=1989c^2</math>, we get <math>\frac{994c*bc\sina}{c^2b\sina}=\boxed{994}</math>. | ||
== See also == | == See also == |
Revision as of 17:51, 21 December 2012
Problem
Let , , be the three sides of a triangle, and let , , , be the angles opposite them. If , find
Solution
Solution 1
We can draw the altitude to , to get two right triangles. , from the definition of the cotangent. From the definition of area, , so .
Now we evaluate the numerator:
From the Law of Cosines and the sine area formula,
Then .
Solution 2
By the Law of Cosines,
Now
Solution 3
Use Law of cosines to give us or therefore . Next, we are going to put all the sin's in term of . We get . Therefore, we get $\cot(\gamma)=\frac{994c}{b\sina}$ (Error compiling LaTeX. Unknown error_msg).
Next, use Law of Cosines to give us . Therefore, . Also, . Hence, .
Lastly, . Therefore, we get .
Now, $\frac{\cot(\gamma)}{\cot(\beta)+\cot(\alpha)}=\frac{\frac{994c}{b\sina}}{\frac{a^2-994c^2+b^2-994c^2}{bc\sin(a)}}$ (Error compiling LaTeX. Unknown error_msg). After using , we get $\frac{994c*bc\sina}{c^2b\sina}=\boxed{994}$ (Error compiling LaTeX. Unknown error_msg).
See also
1989 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 |