# Difference between revisions of "2003 AIME I Problems/Problem 10"

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and so <math>\cos 7^\circ \cos \theta = \sin 7^\circ \sin\theta</math> | and so <math>\cos 7^\circ \cos \theta = \sin 7^\circ \sin\theta</math> | ||

− | and, since <math>0^\circ < \theta < 180^circ</math>, we must have <math>\theta = 83^\circ</math>, so the answer is <math>083</math>. | + | and, since <math>0^\circ < \theta < 180^\circ</math>, we must have <math>\theta = 83^\circ</math>, so the answer is <math>083</math>. |

== See also == | == See also == |

## Revision as of 18:41, 4 November 2006

## Problem

Triangle is isosceles with and Point is in the interior of the triangle so that and Find the number of degrees in

## Solution

From the givens, we have the following angle measures: , . If we define then we also have . Then Apply the Law of Sines to triangles and to get

Clearing denominators, evaluating and applying one of our trigonometric identities to the result gives

and multiplying through by 2 and applying the double angle formula gives

and so

and, since , we must have , so the answer is .