2002 AMC 12P Problems/Problem 25

Problem

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

Given $\begin{cases}\sin{a} + \sin{b} = \frac{\sqrt{2}}{2} \dots \textcircled{1}\\ \cos {a} + \cos {b} = \frac{\sqrt{6}}{2} \dots \textcircled{2}\end{cases}$ We multiply both sides of the syetem, $\textcircled{1} \times \textcircled{2}$ , then we get $(\sin{a}\cos{a} + \sin{b} \cos{b} )+( \sin{a}\cos{b} + \sin{b} \cos{a} )= \frac{\sqrt{3}}{2}$ . i.e. $(\sin{a}\cos{a} + \sin{b} \cos{b} )+\sin{(a+b)}= \frac{\sqrt{3}}{2}$ .

We must get the sum of the first part of the equation, then we calculate $\textcircled{1}^2+\textcircled{2}^2$ , we will get $\sin{a}\cos{a} + \sin{b} \cos{b} = 0$ as $\sin^{2}{a}+\cos^{2}{a} = 1$ and $\sin^{2}{b}+\cos^{2}{b} = 1$ .

So $\sin{(a+b)} = \frac{\sqrt{3}}{2} \Longrightarrow \boxed{\mathrm{C}}$

Comment: This problem is pretty much identical to https://artofproblemsolving.com/wiki/index.php/2007_AMC_12A_Problems/Problem_17 except with different numbers.

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2002 AMC 12P Problems/Problem 25

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