Chứng minh đẳng thức sau sina+sin2a+sin3a+...+sinna
Giải thích
\(\sin a + \sin 2a + \sin 3a + ... + \sin na\)
\( = 2\sin \frac{a}{2}\left( {\sin a + \sin 2a + \sin 3a + ... + \sin na} \right)\)
\( = \left( {\cos \frac{a}{2} - \cos \frac{{3a}}{2}} \right) + \left( {\cos \frac{{3a}}{2} - \cos \frac{{5a}}{2}} \right) + ... + \left[ {\cos \left( {n - \frac{1}{2}} \right)a - \cos \left( {n + \frac{1}{2}} \right)a} \right]\)
\( = \cos \frac{a}{2} - \cos \left( {n + \frac{1}{2}} \right)a\)
\( = \sin \frac{{na}}{2}.\sin \frac{{\left( {n + 1} \right)a}}{2}\)
\( = \frac{{\sin \frac{{na}}{2}.\sin \frac{{\left( {n + 1} \right)a}}{2}}}{{\sin \frac{a}{2}}}\)