Trigonometri Sudut Rangkap

\begin{aligned} \sin2a&=\sin(a+a)\\ &=\sin a\cos a+\cos a\sin a\\ &=\sin a\cos a+\sin a\cos a&{\color{red}\cos a\sin a=\sin a\cos a\text{ (sifat komutatif perkalian)}}\\ &=\boxed{\boxed{2\sin a\cos a}} \end{aligned}

\begin{aligned} \cos2a&=\cos(a+a)\\ &=\cos a\cos a-\sin a\sin a\\ &=\boxed{\boxed{\cos^2 a-\sin^2 a}}\\ &=\cos^2 a-\left(1-\cos^2 a\right)\\ &=\cos^2 a-1+\cos^2 a\\ &=\boxed{\boxed{2\cos^2a-1}} \end{aligned} \begin{aligned} \cos2a&=\cos^2 a-\sin^2 a\\ &=1-\sin^2 a-\sin^2 a\\ &=\boxed{\boxed{1-2\sin^2 a}} \end{aligned}

\begin{aligned} \tan2a&=\tan(a+a)\\ &=\dfrac{\tan a+\tan a}{1-\tan a\cdot\tan a}\\ &=\boxed{\boxed{\dfrac{2\tan a}{1-\tan^2a}}} \end{aligned}

\begin{aligned} \sin3a&=\sin(2a+a)\\ &=\sin 2a\cos a+\cos 2a\sin a\\ &=\left(2\sin a\cos a\right)\cos a+\left(1-2\sin^2a\right)\sin a\\ &=2\sin a\cos^2a+\sin a-2\sin^3a\\ &=2\sin a\left(1-\sin^2a\right)+\sin a-2\sin^3a\\ &=2\sin a-2\sin^3a+\sin a-2\sin^3a\\ &=\boxed{\boxed{3\sin a-4\sin^3a}} \end{aligned}

\begin{aligned} \cos3a&=\cos(2a+a)\\ &=\cos 2a\cos a-\sin 2a\sin a\\ &=\left(2\cos^2a-1\right)\cos a-\left(2\sin a\cos a\right)\sin a\\ &=2\cos^3a-\cos a-2\sin^2 a\cos a\\ &=2\cos^3a-\cos a-2\left(1-\cos^2a\right)\cos a\\ &=2\cos^3a-\cos a-2\cos a+2\cos^3a\\ &=\boxed{\boxed{4\cos^3a-3\cos a}} \end{aligned}

\begin{aligned} \tan3a&=\tan(2a+a)\\ &=\dfrac{\tan 2a+\tan a}{1-\tan 2a\cdot\tan a}\\[3.8pt] &=\dfrac{\dfrac{2\tan a}{1-\tan^2a}+\tan a}{1-\left(\dfrac{2\tan a}{1-\tan^2a}\right)\tan a}\\[3.8pt] &=\dfrac{\dfrac{2\tan a}{1-\tan^2a}+\dfrac{\tan a-\tan^3a}{1-\tan^2a}}{\dfrac{1-\tan^2a}{1-\tan^2a}-\dfrac{2\tan^2 a}{1-\tan^2a}}\\[3.8pt] &=\dfrac{\dfrac{3\tan a-\tan^3a}{1-\tan^2a}}{\dfrac{1-3\tan^2a}{1-\tan^2a}}\\ &=\boxed{\boxed{\dfrac{3\tan a-\tan^3a}{1-3\tan^2a}}} \end{aligned}