Latihan Matematika Peminatan Kelas XI Jumlah, Selisih, dan Perkalian Sinus dan Cosinus
# 10
Pilgan

Nilai dari cos⁡40°−cos⁡50°sin⁡50°−sin⁡40°\frac{\cos40\degree-\cos50\degree}{\sin50\degree-\sin40\degree} adalah ....

A

133\frac{1}{3}\sqrt{3}

B

3\sqrt{3}

C

11

D

−1-1

E

−12-\frac{1}{2}

Pembahasan:

Persoalan di atas dapat diselesaikan dengan menyederhanakan persamaan.

Rumus umum selisih cosinus adalah

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

Rumus umum selisih sinus adalah

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

Dengan demikian,

cos⁡40°−cos⁡50°sin⁡50°−sin⁡40°=−2sin⁡12(40°+50°)sin⁡12(40°−50°)2cos⁡12(50°+40°)sin⁡12(50°−40°)\frac{\cos40\degree-\cos50\degree}{\sin50\degree-\sin40\degree}=\frac{-2\sin\frac{1}{2}\left(40\degree+50\degree\right)\sin\frac{1}{2}\left(40\degree-50\degree\right)}{2\cos\frac{1}{2}\left(50\degree+40\degree\right)\sin\frac{1}{2}\left(50\degree-40\degree\right)}

=−sin⁡12(90°)sin⁡12(−10°)cos⁡12(90°)sin⁡12(10°)=-\frac{\sin\frac{1}{2}\left(90\degree\right)\sin\frac{1}{2}\left(-10\degree\right)}{\cos\frac{1}{2}\left(90\degree\right)\sin\frac{1}{2}\left(10\degree\right)}

=−sin⁡45°sin⁡(−5°)cos⁡45°sin⁡5°=-\frac{\sin45\degree\sin\left(-5\degree\right)}{\cos45\degree\sin5\degree}

Karena sin⁡(−θ)=−sin⁡θ\sin\left(-\theta\right)=-\sin\theta maka

=sin⁡45°sin⁡5°cos⁡45°sin⁡5°=\frac{\sin45\degree\sin5\degree}{\cos45\degree\sin5\degree}

=sin⁡45°cos⁡45°=\frac{\sin45\degree}{\cos45\degree}

=tan⁡45°=\tan45\degree

=1=1

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