2013 AIME II Problems/Problem 15
Let be angles of an acute triangle with There are positive integers , , , and for which where and are relatively prime and is not divisible by the square of any prime. Find .
Let's draw the triangle. Since the problem only deals with angles, we can go ahead and set one of the sides to a convenient value. Let .
By the Law of Sines, we must have and .
Now let us analyze the given:
Now we can use the Law of Cosines to simplify this:
Therefore: Similarly, Note that the desired value is equivalent to , which is . All that remains is to use the sine addition formula and, after a few minor computations, we obtain a result of . Thus, the answer is .
Note that the problem has a flaw because which contradicts with the statement that it's an acute triangle. Would be more accurate to state that and are smaller than 90. -Mathdummy
Let us use the identity .
Add to both sides of the first given equation.
Thus, as we have so is and therefore is .
Similarily, we have and and the rest of the solution proceeds as above.
Adding (1) and (3) we get: or or or
Similarly adding (2) and (3) we get: Similarly adding (1) and (2) we get:
And (4) - (5) gives:
Now (6) - (7) gives: or and so is and therefore is
Now can be computed first and then is easily found.
Thus and can be plugged into (4) above to give x = .
Hence the answer is = .
Let's take the first equation . Substituting for C, given A, B, and C form a triangle, and that , gives us:
Expanding out gives us .
Using the double angle formula , we can substitute for each of the squares and . Next we can use the Pythagorean identity on the and terms. Lastly we can use the sine double angle to simplify.
Expanding and canceling yields, and again using double angle substitution,
Further simplifying yields:
Using cosine angle addition formula and simplifying further yields, and applying the same logic to Equation yields:
Substituting the identity , we get:
Since the third expression simplifies to the expression , taking inverse cosine and using the angles in angle addition formula yields the answer, , giving us the answer .
We will use the sum to product formula to simply these equations. Recall Using this, let's rewrite the first equation: Now, note that . We apply the sum to product formula again. Now, recall that . We apply this and simplify our expression to get: Analogously, We can find this value easily by angle sum formula. After a few calculations, we get , giving us the answer .
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