Determine the sum of angles $A,B,$ where $0^\circ \leq A,B, \leq 180^\circ$ and \[ \sin A + \sin B = \sqrt{\frac{3}{2}}, \cos A + \cos B = \sqrt{\frac{1}{2}} \]
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Tags: trigonometry, LaTeX
28.01.2006 01:29
Thanks for the help! This is the last problem on COMC 1996 Part I by the way.
28.01.2006 01:42
I think I have a method. Let me work it out on paper to see if it works. Edit: It dodn't really work. I'll think about this more later if nobody posts a solution.
28.01.2006 02:32
28.01.2006 08:59
Having practised on this question myself, it's pretty hard to forget the solution
28.01.2006 09:54
The Artzilla's proof. $\sin A+\sin B=\sqrt{\frac 32},\ \cos A+\cos B=\sqrt{\frac 12}\ .$ $\frac{\sin A+\sin B}{\cos A+\cos B}=\frac{2\sin \frac{A+B}{2}\cos \frac{A-B}{2}}{2\cos \frac{A+B}{2}\cos \frac{A-B}{2}}=\tan \frac{A+B}{2}\ .$ $\Longrightarrow \tan \frac{A+B}{2}=\sqrt 3\Longrightarrow A+B=120^{\circ}\ .$
28.01.2006 21:13
28.01.2006 21:28
try squaring both equations and then adding them together. it should be pretty obvious from there.
29.01.2006 00:14
29.01.2006 03:29
Thanks guys! I liked all three solutions! P.S. 1234567890, you have typo in the first one. I assume you meant on the first line as $z_1 = \cos A + i \sin A = \text{cis} A$. You forgot $\sin A$ part.