Sin(1) + sin (2) + sin (3) + .........sin(89)
Anonymous:
is the question sin^2 (1)..+sin^2 (2)...
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Y = Sin 1° + sin 2° + sin 3° ... + sin 88° + sin 89°
Multiply both sides with 2 sin 1°.
2 Y sin 1° = 2 sin 1° sin 1° + 2 sin 2° sin 1° + 2 sin 3 sin 1°+ ...
+ 2 sin 88° sin 1° + 2 sin 89 sin 1°
= cos 0° - cos 2° + cos 1° - cos 3° + cos 2° - cos 4° + cos 3° - cos 5...
... + cos 86 - cos 88° + cos 87° - cos 89° + cos 88° - cos 90°
= 1 + cos 1° - cos 89°
Y = [1 + cos 1° - sin 1°] / (2 sin 1°) = 1/2 * [ cot 1/2° - 1]
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we can do this by using complex numbers using De Moivre's formula:
Let 1° = 1*π/180 rad = a
![Y=Imaginary\ part\ of\ \Sigma_a^{89a}{e^{ia}}\\\\=Im[ \frac{e^{ia}(1-e^{89a})}{1-e^{ia}} ]\\\\=Im[ \frac{e^{ia}-e^{i90a}}{1-e^{ia}}]=IM[ \frac{cos\ a-i\ (1-sin\ a)}{1-cos\ a + i\ sin a} ]\\\\=Im[ \frac{cos\ a + sin\ a -1}{2(1-cos\ a)}*(1 +i) ]\\\\=\frac{1}{2}(cot\ 1^o-1) Y=Imaginary\ part\ of\ \Sigma_a^{89a}{e^{ia}}\\\\=Im[ \frac{e^{ia}(1-e^{89a})}{1-e^{ia}} ]\\\\=Im[ \frac{e^{ia}-e^{i90a}}{1-e^{ia}}]=IM[ \frac{cos\ a-i\ (1-sin\ a)}{1-cos\ a + i\ sin a} ]\\\\=Im[ \frac{cos\ a + sin\ a -1}{2(1-cos\ a)}*(1 +i) ]\\\\=\frac{1}{2}(cot\ 1^o-1)](https://tex.z-dn.net/?f=Y%3DImaginary%5C+part%5C+of%5C+%5CSigma_a%5E%7B89a%7D%7Be%5E%7Bia%7D%7D%5C%5C%5C%5C%3DIm%5B+%5Cfrac%7Be%5E%7Bia%7D%281-e%5E%7B89a%7D%29%7D%7B1-e%5E%7Bia%7D%7D+%5D%5C%5C%5C%5C%3DIm%5B+%5Cfrac%7Be%5E%7Bia%7D-e%5E%7Bi90a%7D%7D%7B1-e%5E%7Bia%7D%7D%5D%3DIM%5B+%5Cfrac%7Bcos%5C+a-i%5C+%281-sin%5C+a%29%7D%7B1-cos%5C+a+%2B+i%5C+sin+a%7D+%5D%5C%5C%5C%5C%3DIm%5B+%5Cfrac%7Bcos%5C+a+%2B+sin%5C+a+-1%7D%7B2%281-cos%5C+a%29%7D%2A%281+%2Bi%29+%5D%5C%5C%5C%5C%3D%5Cfrac%7B1%7D%7B2%7D%28cot%5C+1%5Eo-1%29)
Multiply both sides with 2 sin 1°.
2 Y sin 1° = 2 sin 1° sin 1° + 2 sin 2° sin 1° + 2 sin 3 sin 1°+ ...
+ 2 sin 88° sin 1° + 2 sin 89 sin 1°
= cos 0° - cos 2° + cos 1° - cos 3° + cos 2° - cos 4° + cos 3° - cos 5...
... + cos 86 - cos 88° + cos 87° - cos 89° + cos 88° - cos 90°
= 1 + cos 1° - cos 89°
Y = [1 + cos 1° - sin 1°] / (2 sin 1°) = 1/2 * [ cot 1/2° - 1]
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we can do this by using complex numbers using De Moivre's formula:
Let 1° = 1*π/180 rad = a
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