Given-sin-a-sin-b-2-2-cos-a-cos-b-6-2-for-a-b-real-numbers-Evaluate-sin-a-b-A-3-2-D-3-2-B-2-3-E-2-3-C-3-4-

Question Number 176581 by cortano1 last updated on 22/Sep/22

Given {\begin{cases} \sin a + \sin b = \frac{\sqrt{2}}{2} \\ \cos a + \cos b = \frac{\sqrt{6}}{2} \end{cases}

for a, b real numbers . Evaluate

\sin (a + b) .

(A) \frac{\sqrt{3}}{2} (D) - \frac{\sqrt{3}}{2}

(B) \frac{2}{\sqrt{3}} (E) - \frac{2}{\sqrt{3}}

(C) \frac{\sqrt{3}}{4}

Answered by som(math1967) last updated on 22/Sep/22

\frac{\sin a + s i n b}{c o s a + c o s b} = \frac{\sqrt{2}}{\sqrt{6}}

\Rightarrow t a n (\frac{a + b}{2}) = \frac{1}{\sqrt{3}}

s i n (a + b) = \frac{2 . \frac{1}{\sqrt{3}}}{1 + {(\frac{1}{\sqrt{3}})}^{2}}

= \frac{2}{\sqrt{3}} \times \frac{3}{4} = \frac{\sqrt{3}}{2} \dots \dots (A)