Difference between revisions of "2002 AMC 12P Problems/Problem 25"

Revision as of 01:05, 30 December 2023

Problem

Let $a$ and $b$ be real numbers such that $\sin{a} + \sin{b} = \sqrt{2}{2}$ and $\cos {a} + \cos {b} = \sqrt{6}{2}.$ Find $\sin{(a+b)}.$

$\text{(A) }\frac{1}{2} \qquad \text{(B) }\frac{\sqrt{2}}{2} \qquad \text{(C) }\frac{\sqrt{3}}{2} \qquad \text{(D) }\frac{\sqrt{6}}{2} \qquad \text{(E) }1$

Solution

If $\log_{b} 729 = n$ , then $b^n = 729$ . Since $729 = 3^6$ , $b$ must be $3$ to some factor of 6. Thus, there are four (3, 9, 27, 729) possible values of $b \Longrightarrow \boxed{\mathrm{E}}$ .

2002 AMC 12P (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 1: / Line 1: @@
 == Problem ==
-How many positive [[integer]]s <math>b</math> have the property that <math>\log_{b} 729</math> is a positive integer?
+Let <math>a</math> and <math>b</math> be real numbers such that <math>\sin{a} + \sin{b} = \sqrt{2}{2}</math> and <math>\cos {a} + \cos {b} = \sqrt{6}{2}.</math> Find <math>\sin{(a+b)}.</math>
-<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 }  </math>
+<math>
+\text{(A) }\frac{1}{2}
+\qquad
+\text{(B) }\frac{\sqrt{2}}{2}
+\qquad
+\text{(C) }\frac{\sqrt{3}}{2}
+\qquad
+\text{(D) }\frac{\sqrt{6}}{2}
+\qquad
+\text{(E) }1
+</math>
 == Solution ==

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Difference between revisions of "2002 AMC 12P Problems/Problem 25"

Revision as of 01:05, 30 December 2023

Problem

Solution

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