Latihan Matematika Peminatan Kelas XI Jumlah, Selisih, dan Perkalian Sinus dan Cosinus
# 7
Pilgan

Nilai dari cos⁡210°−cos⁡30°+sin⁡45°\cos210\degree-\cos30\degree+\sin45\degree adalah ....

A

123−2\frac{1}{2}\sqrt{3}-\sqrt{2}

B

122−3\frac{1}{2}\sqrt{2}-\sqrt{3}

C

12(2−3)\frac{1}{2}\left(\sqrt{2}-\sqrt{3}\right)

D

12(3−2)\frac{1}{2}\left(\sqrt{3}-\sqrt{2}\right)

E

3−2\sqrt{3}-\sqrt{2}

Pembahasan:

Rumus umum selisih cosinus adalah

cos⁡α−cos⁡β=−2(sin⁡12(α+β)sin⁡12(α−β))\cos\alpha-\cos\beta=-2\left(\sin\frac{1}{2}\left(\alpha+\beta\right)\sin\frac{1}{2}\left(\alpha-\beta\right)\right)

Dengan demikian, hitung terlebih dahulu selisih cosinus

cos⁡210°−cos⁡30°=−2(sin⁡12(210°+30°)sin⁡12(210°−30°))\cos210\degree-\cos30\degree=-2\left(\sin\frac{1}{2}\left(210\degree+30\degree\right)\sin\frac{1}{2}\left(210\degree-30\degree\right)\right)

=−2(sin⁡12(240°)sin⁡12(180°))=-2\left(\sin\frac{1}{2}\left(240\degree\right)\sin\frac{1}{2}\left(180\degree\right)\right)

=−2(sin⁡120°sin⁡90°)=-2\left(\sin120\degree\sin90\degree\right)

=−2(123 . 1)=-2\left(\frac{1}{2}\sqrt{3}\ .\ 1\right)

=−3=-\sqrt{3}

Selanjutnya hasil selisih dijumlahkan dengan sin⁡45°\sin45\degree

=−3+sin⁡45° =-\sqrt{3}+\sin45\degree\

=−3+122=-\sqrt{3}+\frac{1}{2}\sqrt{2}

=122−3=\frac{1}{2}\sqrt{2}-\sqrt{3}

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