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SMUP 2010

Solusi SMUP 2010 Kode 041 – Nomor 22


Hasil\ dari\ sin^{2}\ 1^\circ +sin^{2}\ 2^\circ +sin^{2}\ 3^\circ + .... +sin^{2}\ 89^\circ \ adalah\ ....
A.\quad  43\\B.\quad 43,5\\C.\quad 44\\D.\quad 44,5\\E.\quad 90
Jawaban: D

sin\ \alpha ^\circ =cos\ (90-\alpha )^\circ
sin\ 89^\circ =cos\ 1^\circ \quad \rightarrow \quad sin^{2}\ 89^\circ =cos^{2}\ 1^\circ \\sin\ 88^\circ =cos\ 2^\circ \quad \rightarrow \quad sin^{2}\ 88^\circ =cos^{2}\ 2^\circ \\sin\ 87^\circ =cos\ 3^\circ \quad \rightarrow \quad sin^{2}\ 87^\circ =cos^{2}\ 3^\circ \\....
sedangkan:

sin^{2}\ \alpha ^\circ +cos^{2}\ \alpha ^\circ =1
sehingga:

sin^{2}\ 1^\circ +sin^{2}\ 2^\circ +sin^{2}\ 3^\circ + .... +sin^{2}\ 89^\circ \\\\=\big(sin^{2}\ 1^\circ +sin^{2}\ 89^\circ \big)+\big(sin^{2}\ 2^\circ +sin^{2}\ 88^\circ \big)+\big(sin^{2}\ 3^\circ +sin^{2}\ 87^\circ \big)+ .... +\big(sin^{2}\ 44^\circ +sin^{2}\ 46^\circ \big)+sin^{2}\ 45^\circ \\\\=\underbrace{\big(sin^{2}\ 1^\circ +cos^{2}\ 1^\circ \big)}_{1}+\underbrace{\big(sin^{2}\ 2^\circ +cos^{2}\ 2^\circ \big)}_{1}+\underbrace{\big(sin^{2}\ 3^\circ +cos^{2}\ 3^\circ \big)}_{1}+ .... +\underbrace{\big(sin^{2}\ 44^\circ +cos^{2}\ 44^\circ \big)}_{1}+\Big(\frac{1}{2}\sqrt{2}\Big)^{2}\\\\=44+\big(\frac{1}{4}\times 2\big)\\\\=44,5

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