Latihan Perkalian Fungsi Trigonometri

\(16\sqrt{2}\cdot \sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ}\)

\(-8\sqrt{2}\cdot \color{purple}(-2\sin 10^{\circ} \sin 70^{\circ})\color{black}\sin 50^{\circ}\)

\(-8\sqrt{2}\cdot \color{purple}[\cos (10^{\circ} + 70^{\circ} )\:-\: \cos (10^{\circ} \:-\: 70^{\circ} )]\color{black}\sin 50^{\circ}\)

\(-8\sqrt{2}\cdot \color{purple}[\cos 80^{\circ}\:-\: \cos (-60)^{\circ} ]\color{black}\sin 50^{\circ}\)

\(-8\sqrt{2}\cdot \color{purple}(\cos 80^{\circ}\:-\: \dfrac{1}{2} )\color{black}\sin 50^{\circ}\)

\((-8\sqrt{2}\cos 80^{\circ} + 4\sqrt{2})\cdot \sin 50^{\circ}\)

\(-8\sqrt{2}\cos 80^{\circ}\cdot  \sin 50^{\circ} + 4\sqrt{2}\cdot \sin 50^{\circ}\)

\(-4\sqrt{2}\cdot \color{purple}2 \cdot \cos 80^{\circ}\cdot  \sin 50^{\circ} \color{black}+ 4\sqrt{2}\cdot \sin 50^{\circ}\)

\(-4\sqrt{2} [\sin (80^{\circ} + 50^{\circ}) \:-\: \sin (80^{\circ} \:-\: 50^{\circ})] + 4\sqrt{2}\cdot \sin 50^{\circ}\)

\(-4\sqrt{2} (\sin 130^{\circ} \:-\: \sin 30^{\circ}) + 4\sqrt{2}\cdot \sin 50^{\circ}\)

\(-4\sqrt{2} (\sin 130^{\circ} \:-\: \dfrac{1}{2}) + 4\sqrt{2}\cdot \sin 50^{\circ}\)

\(-4\sqrt{2} \sin 130^{\circ} + 2\sqrt{2} + 4\sqrt{2}\cdot \sin 50^{\circ}\)

\(-4\sqrt{2} \sin (180^{\circ}\:-\: 50^{\circ})  + 2\sqrt{2} + 4\sqrt{2}\cdot \sin 50^{\circ}\)

\(-4\sqrt{2} \sin 50^{\circ}+ 2\sqrt{2} + 4\sqrt{2}\cdot \sin 50^{\circ}\)

\(2\sqrt{2}\)

\([\cos \dfrac{\pi}{7}\:-\:\cos \dfrac {2\pi}{7} + \cos \dfrac {3\pi}{7}]\times \color{red}\dfrac{2\sin \dfrac{2\pi}{7}}{2\sin \dfrac{2\pi}{7}}\)

\(\dfrac{2\sin \dfrac{2\pi}{7}\cos \dfrac{\pi}{7}\:-\: 2\sin \dfrac{2\pi}{7}\cos \dfrac{2\pi}{7} + 2\sin \dfrac{2\pi}{7}\cos \dfrac{3\pi}{7}}{2\sin \dfrac{2\pi}{7}}\)

\(\dfrac{\sin (\dfrac{2\pi}{7} + \dfrac{\pi}{7}) + \sin (\dfrac{2\pi}{7}\:-\: \dfrac{\pi}{7}) – [\sin (\dfrac{2\pi}{7} + \dfrac{2\pi}{7}) + \sin (\dfrac{2\pi}{7}\:-\: \dfrac{2\pi}{7})] + \sin (\dfrac{2\pi}{7} + \dfrac{3\pi}{7}) + \sin (\dfrac{2\pi}{7}\:-\: \dfrac{3\pi}{7})          }{2\sin \dfrac{2\pi}{7}}\)

\(\dfrac{\sin \dfrac{3\pi}{7} + \sin \dfrac{\pi}{7} \:-\: (\sin \dfrac{4\pi}{7} + 0) + \sin \dfrac{5\pi}{7} \:-\: \sin \dfrac{\pi}{7}}{2\sin \dfrac{2\pi}{7}}\)

\(\dfrac{\sin \dfrac{3\pi}{7}\:-\:\sin \dfrac{4\pi}{7} + \sin \dfrac{5\pi}{7}}{2\sin \dfrac{2\pi}{7}}\)

\(\dfrac{2\cos \dfrac{1}{2}\left(\dfrac{3\pi}{7} +\dfrac{4\pi}{7}\right)\sin \dfrac{1}{2}\left(\dfrac{3\pi}{7}\:-\:\dfrac{4\pi}{7}\right) + \sin \dfrac{5\pi}{7}}{2\sin \dfrac{2\pi}{7}}\)

\(\dfrac{-2\cos \dfrac{1}{2}\pi \cdot\sin \dfrac{\pi}{14}+ \sin \dfrac{5\pi}{7}}{2\sin \dfrac{2\pi}{7}}\)

\(\dfrac{0+ \sin \dfrac{5\pi}{7}}{2\sin \dfrac{2\pi}{7}}\)

\(\dfrac{0+ \sin \left(\pi\:-\: \dfrac{2\pi}{7}\right)}{2\sin \dfrac{2\pi}{7}}\)

\(\dfrac{\cancel{\sin \dfrac{2\pi}{7}}}{2\cancel{\sin \dfrac{2\pi}{7}}}\)

\(\dfrac{1}{2}\)

Menghitung bagian I:

\(\color{blue}\sin \text{A}\cdot \cos \text{B} = \frac{1}{2}[\sin \text{(A + B)} + \sin \text{(A − B)}]\)

\(\sin \frac{13\pi}{12}\cdot \cos \frac{\pi}{12}\)

\(\frac{1}{2}[\sin (\frac{13\pi}{12} + \frac{\pi}{12}) + \sin (\frac{13\pi}{12}\:-\:\frac{\pi}{12})]\)

\(\frac{1}{2}[\sin \frac{14\pi}{12} + \sin\frac{12\pi}{12}]\)

\(\frac{1}{2}[\sin \frac{7\pi}{6} + \sin \pi ]\)

\(\frac{1}{2}[-\frac{1}{2} + 0 ]\)

\(-\frac{1}{4}\)

Menghitung bagian II:

\(\color{blue}-2\sin \text{A}\cdot \sin \text{B} = \cos\text{(A + B)}\:-\:\cos \text{(A − B)}\)

\(2\sin \frac{7\pi}{12}\cdot \sin \frac{\pi}{12}\)

\(-[\cos (\frac{7\pi}{12} + \frac{\pi}{12})\:-\:\cos (\frac{7\pi}{12}\:-\:\frac{\pi}{12})]\)

\(-[\cos \frac{8\pi}{12}\:-\:\cos \frac{6\pi}{12}]\)

\(-\cos \frac{2\pi}{3} + \cos \frac{\pi}{2}\)

\(-(-\frac{1}{2}) + 0\)

\(\frac{1}{2}\)

Jadi, \(\sin \frac{13\pi}{12}\cdot \cos \frac{\pi}{12} + 2\sin \frac{7\pi}{12}\cdot \sin \frac{\pi}{12} = -\frac{1}{4} + \frac{1}{2} = \frac{1}{4}\)

\(\color{blue} \cos \text{2A} = 2\cos^2 \text{A}\:-\:1\)

\(\color{blue} \cos^2 \text{A} = \frac{\cos \text{2A}}{2} + \frac{1}{2}\)

\(\color{blue} \sin\text{2A}  = 2\sin \text{A}\cdot\cos \text{A}\)

\(\color{blue} \cos \text{A}\cdot \cos \text{B} = \frac{1}{2}[\cos \text{(A + B)} + \cos \text{(A − B)}]\)

\(\dfrac{(\cos^2 10^{\circ}\:-\:\frac{1}{2})(\cos^2 20^{\circ}\:-\:\frac{1}{2})(\cos^2 40^{\circ}\:-\:\frac{1}{2})}{\sin 15^{\circ}\cdot \cos 15^{\circ}}\)

\(\dfrac{(\frac{\cos 20^{\circ}}{2} + \frac{1}{2}\:-\:\frac{1}{2})(\frac{\cos 40^{\circ}}{2} + \frac{1}{2}\:-\:\frac{1}{2})(\frac{\cos 80^{\circ}}{2} + \frac{1}{2}\:-\:\frac{1}{2})}{\frac{1}{2}\sin 30^{\circ}}\)

\(\dfrac{(\frac{\cos 20^{\circ}}{2})(\frac{\cos 40^{\circ}}{2})(\frac{\cos 80^{\circ}}{2})}{\frac{1}{4}}\)

\(\frac{1}{2}\cdot \frac{1}{2} \cdot \frac{1}{2}\cdot \frac{4}{1}\cdot \cos 20^{\circ}\cdot \cos 40^{\circ} \cdot \cos 80^{\circ}\)

\(\frac{1}{2}\cdot \cos 40^{\circ}\cdot \cos 80^{\circ} \cdot \cos 20^{\circ}\)

\(\frac{1}{2}\cdot \frac{1}{2}[\cos 120^{\circ} + \cos (-40^{\circ})] \cdot \cos 20^{\circ}\)

\(\frac{1}{4}[-\frac{1}{2} + \cos 40^{\circ}]\cdot \cos 20^{\circ}\)

\(-\frac{1}{8}\cos 20^{\circ} + \frac{1}{4}\cos 40^{\circ}\cos 20^{\circ}\)

\(-\frac{1}{8}\cos 20^{\circ} + \frac{1}{4}\cdot\frac{1}{2}[\cos 60^{\circ} + \cos (-20^{\circ})]\)

\(-\frac{1}{8}\cos 20^{\circ} + \frac{1}{8}[\frac{1}{2} + \cos 20^{\circ}]\)

\(-\frac{1}{8}\cos 20^{\circ} + \frac{1}{16} + \frac{1}{8}\cos 20^{\circ}]\)

\(\frac{1}{16}\)

\(4\cdot 4 \cdot \sin^2x \cdot \cos^2 x \cdot \cos x\)

\(4 \cos x \cdot 4 \sin^2x \cos^2 x\)

\(4 \cos x (2 \sin x \cos x)^2\)

\(4 \cos x \cdot \sin^2 {2x}\)

\(2 \cdot \cos x \sin {2x} \cdot \sin {2x}\)

\(2 [\sin (x + 2x) \:-\:\sin (x \:-\:2x) ] \cdot \sin 2x\)

\(2 [\sin 3x + \sin x] \cdot \sin 2x\)

\(2\sin 3x \sin 2x + 2\sin x \sin 2x\)

\(-(\cos 5x \:-\:\cos x) \:-\:(\cos 3x \:-\:\cos x)\)

\(-\cos 5x + \cos x \:-\:\cos 3x + \cos x\)

\(2\cos x \:-\:\cos 5x \:-\:\cos 3x\)

Terbukti!