Sin90×cos0+ sin88×cos 2+sin 86×cos 4+......+sin2×cos88+ sin0×cos90
Answers
Explanation:
= 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
\begin{lgathered}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)\end{lgathered}
Y=Imaginary part of Σ
a
89a
e
ia
=Im[
1−e
ia
e
ia
(1−e
89a
)
]
=Im[
1−e
ia
e
ia
−e
i90a
]=IM[
1−cos a+i sina
cos a−i (1−sin a)
]
=Im[
2(1−cos a)
cos a+sin a−1
∗(1+i)]
=
2
1
(cot 1
o
−1)