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Nilai eksak dari (cos 40° + cos 20°) × (cos 120° + cos 20°) adalah 3/4.

Penjelasan dengan langkah-langkah:

Kita akan menentukan nilai eksak dari operasi fungsi trigonometri:
(cos 40° + cos 20°) × (cos 120° + cos 20°).

Penyelesaian:

\begin{aligned}&\left(\cos40^{\circ}+\cos20^{\circ}\right)\left(\cos120^{\circ}+\cos20^{\circ}\right)\\&\quad\rightarrow \cos a+\cos b=2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)\\&{=\ }\left[2\cos\left(\frac{40^{\circ}+20^{\circ}}{2}\right)\cos\left(\frac{40^{\circ}-20^{\circ}}{2}\right)\right]\left[\cos\left(90^{\circ}+30^{\circ}\right)+\cos20^{\circ}\right]\\&\quad\rightarrow \cos\left(90^{\circ}+a\right)=-\sin a\end{aligned}
\begin{aligned}&{=\ }\left(2\cos30^{\circ}\cos10^{\circ}\right)\left(-\sin30^{\circ}+\cos20^{\circ}\right)\\&{=\ }\left(2\cdot\frac{\sqrt{3}}{2}\cdot\cos10^{\circ}\right)\left(-\frac{1}{2}+\cos20^{\circ}\right)\\&{=\ }\left(\sqrt{3}\cdot\cos10^{\circ}\right)\left(-\frac{1}{2}+\cos20^{\circ}\right)\\&{=\ }\frac{\sqrt{3}}{2}\left(-\cos10^{\circ}+2\cos20^{\circ}\cos10^{\circ}\right)\\\end{aligned}
\begin{aligned}&{=\ }\frac{\sqrt{3}}{2}\left(-\cos10^{\circ}+2\cos\left(\frac{30^{\circ}+10^{\circ}}{2}\right)\cos\left(\frac{30^{\circ}-10^{\circ}}{2}\right)\right)\\&{=\ }\frac{\sqrt{3}}{2}\left(\cancel{-\cos10^{\circ}}+\cos30^{\circ}+\cancel{\cos10^{\circ}}\right)\\&{=\ }\frac{\sqrt{3}}{2}\cdot\cos30^{\circ}\\&{=\ }\frac{\sqrt{3}}{2}\cdot\frac{\sqrt{3}}{2}\\&{=\ }\boxed{\,\bf\frac{3}{4}\,}\end{aligned}
\blacksquare

  • henriyulianto
    sama2
  • Martin1103
    Terima kasih pak

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