A-maths

a) Let P ( n ) be the proposition “ sin x+sin2x+sin3x+...+sin nx =sin(1/2)(n+1)x sin(n/2)x / sin (x/2) where sinx/2不等於0”.

When n = 1,

L.H.S. = sin x

R.H.S. = sin(1/2)(1+1)x sin(1/2)x / sin (x/2)

= sin x

= L.H.S.

Therefore P ( 1 ) is true.

Assume P ( k ) is true for some positive integers k.

i.e. sin x+sin2x+sin3x+...+sin kx =sin(1/2)(k+1)x sin(k/2)x / sin (x/2)

When n = k + 1 ,

L.H.S. = sin x+sin2x+sin3x+...+sin kx + sin ( k + 1 )x

= {sin(1/2)(k+1)x sin(k/2)x / sin (x/2)} + sin ( k + 1 )x

= {sin(1/2)(k+1)x sin(k/2)x + sin ( k + 1 )x sin (x/2) } / sin (x / 2)

= {sin(1/2)(k+1)x sin (k/2)x + 2 sin ½ ( k + 1 )x cos ½ ( k + 1 )x sin ( x / 2 ) } / sin ( x / 2 )

= sin ½ ( k + 1 )x {sin (k/2)x + 2 cos ½ ( k + 1 )x sin ( x / 2 ) } / sin ( x / 2 )

= sin ½ ( k + 1 )x {sin (k/2)x + sin ½ ( 1 + k + 1 )x + sin ½ ( 1 – k – 1 )x } / sin ( x / 2 )

= sin ½ ( k + 1 )x {sin (k/2)x + sin ½ ( k + 2 )x - sin (k/2)x } / sin ( x / 2 )

= sin ½ ( k + 1 )x sin ½ ( k + 2 )x / sin ( x / 2 )

R.H.S. = sin ½ ( k + 1 )x sin ½ ( k + 1 + 1 )x / sin ( x / 2 )

= sin ½ ( k + 1 )x sin ½ ( k + 2 )x / sin ( x / 2 )

= L.H.S.

Therefore P ( k + 1 ) is true.

By the principal of mathematical induction, P( n ) is true for all positive integers n where sinx/2不等於0.

b) sinx+sin2x+sin3x+sin4x+sin5x=0

sin(1/2)(5+1)x sin(5/2)x / sin (x/2) = 0

sin 3x sin (5/2) x / sin (x/2) = 0

sin 3x = 0 or sin ( 5/2)x = 0

3x = 180* or ( 5 / 2 ) x = 180*

x = 60* or x = 72* where 0 < x < 90*