[tex]sin^2(1)+sin^2(2)+......+sin^2(90)=?[/tex]
O sa ne folosim de proprietatea fundamentala a trigonometriei ,care este:
[tex](sin x)^2+(cos x)^2=1[/tex]
Alta proprietate : [tex]sin(90-x)=cos(x)[/tex]
Inlocuim cos cu sin in proprietatea fundamentala:
[tex](sin x)^2+(sin(90-x))^2=1[/tex]
Si vom folosi aceasta relatie,grupand cate doi termeni in felul urmator :
[tex](sin 1)^2+(sin 2)^2+...+(sin 88)^2+(sin 89)^2+(sin 90)^2= \\
\{(sin 1)^2+(sin 89)^2)\}+\{(sin 2 )^2+( sin 88)^2\}+.....+ \\
+\{(sin 44)^2+(sin 46)^2\}+\{(sin 45)^2+(sin 90)^2\}= \\
1+1+....+1+(sin 45)^2+1=45+ \frac{1}{2} =91/2[/tex]
