2005 ce maths mc 46

sin^1+sin^2 3 + sin^2 5+...+sin^2 87 + sin^2 89

=(sin^2 1 + sin^2 89)+(sin^2 3+sin^2 87)+...+(sin^2 43+sin^2 47)+sin^2 45


=[sin^2 1+cos^2(90-89)}+[sin^2 3+cos^2(90-87)+...+[sin^2 43+cos^2 (90-47)]+(開方2 /2)^2

=(sin^2 1+cos^2 1)+(sin^2 3+cos^2 3 )+...+(sin^2 43+cos^2 43)=1/2

=1+1+...+1+1/2

=22+1/2
=22.5

sin^2 1 + sin^2 3 +sin^2 5 + ...+ sin^2 87 + sin^2 89
= sin^2 1 + sin^2 89 + ... + sin^2 45
= sin^2 1 + sin^2 (90-1) + ... + sin^2 45
= sin^2 1 + cos^2 1 + ... + sin^2 45
= 1 + sin^2 3 + sin^2 87 + ... + sin^2 45
= 1 + 1 + ... + sin^2 45
= 23 + sin^2 45
= 23 + (1/sqt(2))^2
= 23 + 1/2
= 23.5

sin^2 1 + sin^2 3 +sin^2 5 + ...+ sin^2 87 + sin^2 89
=sin^2 1+sin^2 3 +sin^2 5 +...+sin^2 43 + sin^2 45+sin^2 47 + ...+sin^2 89
=sin^2 1+sin^2 3 +sin^2 5 +...+sin^2 43 + sin^2 45+(sin^2 (90-43) + ...+sin^2 (90-1)
=sin^2 1+sin^2 3 +sin^2 5 +...+sin^2 43 + sin^2 45+cos^2 43 + ...+cos^2 1
=1+1+1+...+1(42 terms) + sin^2 45
=42+4/2
=42+2
=44

2008-03-25 11:25:57 補充：
sorry there is some mistake on the top
after
=sin^2 1+sin^2 3 +sin^2 5 +...+sin^2 43 + sin^2 45+cos^2 43 + ...+cos^2 1
=1+1+1+...+1(22 terms) + sin^2 45
=22+4/2
=22+2
=24

2008-03-25 11:29:09 補充：
sorry mistake again
=1+1+1+...+1(22 terms) + sin^2 45
=22+2/4
=22+0.5
=22.5