Question

sin^2(1) + sin^2(2) + ... + sin^2(90) = 91/2

Answer 1

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Answer 2

$sin^2(1)+sin^2(2)+......+sin^2(90)=?$

O sa ne folosim de proprietatea fundamentala a trigonometriei ,care este:
$(sin x)^2+(cos x)^2=1$

Alta proprietate : $sin(90-x)=cos(x)$
Inlocuim cos cu sin in proprietatea fundamentala:
$(sin x)^2+(sin(90-x))^2=1$
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]