find-the-sum-of-sin-2-1-sin-2-2-sin-2-60-

Question Number 209706 by mr W last updated on 19/Jul/24

f i n d t h e s u m o f

\sin^{2} 1 ° + \sin^{2} 2 ° + \dots + \sin^{2} 60 ° = ?

Answered by BaliramKumar last updated on 19/Jul/24

\underset{x = 1}{\sum^{60}} \sin^{2} x = \underset{x = 1}{\sum^{29}} \sin^{2} x + 15.5

Commented by mr W last updated on 19/Jul/24

a n d \underset{x = 1}{\sum^{29}} \sin^{2} x = ?

Answered by MM42 last updated on 19/Jul/24

{s i n}^{2} a = \frac{1}{2} (1 - c o s 2 a)

\Rightarrow S = 30 - \frac{1}{2} (c o s 2 + c o s 4 + \dots + c o s 120)

s = c o s 2 + c o s 4 + \dots + c o s 120

\Rightarrow 2 s i n 1 \times s = 2 s i n 1 c o s 2 + 2 s i n 1 c o s 4 + \dots + 2 s i n 1 c o s 120

= (s i n 3 - s i n 1) + (s i n 5 - s i n 3) + \dots + (s i n 121 - s i n 119)

= s i n 121 - s i n 1 = 2 s i n 60 c o s 61

\Rightarrow s = \frac{\sqrt{3}}{2} \times \frac{c o s 61}{s i n 1} \sim 24.06

\Rightarrow S \sim 30 - 12.03 = 17.7 ✓

Commented by mr W last updated on 19/Jul/24

t h a n k s s i r!

p l e a s e r e c h e c k .

S = 30 - \frac{\sqrt{3} \cot 1 ° - 3}{8} \approx 17.97

Commented by MM42 last updated on 19/Jul/24

c o s 61 = c o s 60 c o s 1 - s i n 1 s i n 60

= \frac{1}{2} (c o s 1 - \sqrt{3} s i n 1)

\div 2 s i n 1 = \frac{1}{4} (c o t 1 - \sqrt{3}) ★

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