Prove inequality

Dec 2015
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169
Earth
If \(\displaystyle 0<a_1 < a_2 < a_3 <... <a_n < \frac{\pi}{2}\) .
Prove that \(\displaystyle \tan(a_1 ) <\frac{sin(a_1) +sin(a_2 ) +sin(a_3)+...+sin(a_n )}{cos(a_1 ) +cos(a_2 )+cos(a_3 )+...+cos(a_n )}<\tan(a_n )\) .
 
Jun 2019
493
262
USA
If you take unit vectors $e^{ia_j}$, the central term is the tangent of the angle of the vector sum ($Im(\Sigma e^{ia_j}) / Re(\Sigma e^{ia_j})$). The angle of a vector sum is the weighted average of the angles of the vectors. With unit vectors, the central term is $tan \left( \frac{a_1 + a_2 + … + a_n}{n} \right)$.
 
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Mar 2015
182
68
Universe 2.71828i3.14159
Prove it for n=1 and n=2. Then use induction.
DarnItJimImAnEngineer's way is the better one.
 
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