Question

sin²( 5 + 10 +15 + 20 + 25 + 30 + 35 +40 + 45 + 50 + 55 + 60 + 65 + 70 + 75 + 80 + 85 + 90)

Answer 1

Sin² (5+10+15+20+25+30+35+40+45+50+55+60+65+70+75+80+85+90)
= sin² [ (5+85) + (10+80) + (15+75) + (20+70) + (25+65) + (30+60) + (35+55) + (40+50) + 45+90 ]
= sin² [ 90 + 90 + 90 + 90 + 90 + 90 + 90 + 90 + 45 + 90 ]
= sin² [ 855 ]
= sin² [720 + 135 ]
= sin² (135)

= $( \frac{1}{ \sqrt{2} } )^{2}$

= $\frac{1}{2}$

Answer 2

The angle θ is an arithmetic series with 90/5 = 19 terms and common difference d = 5 and first term = 5.

$\theta = (5+90)/2 * (90/5) = 19 * 45 = 20 * 45 - 45 = 5\pi - \pi/4\\ \\Sin^2(5 \pi - \pi /4) = Sin^2 \pi /4 = (1/\sqrt2)^2 = 1/2$