Rumus Jumlah dan Selisih Dua Sudut Sinus dan Kosinus (Kelas XI MIPA)

A. Rumus Jumlah dan Selisih Dua Sudut

\color{blue}\begin{cases} \sin \left ( \alpha +\gamma \right ) & =\sin \alpha \cos \gamma +\cos \alpha \sin \gamma \\ \color{green}\sin \left ( \alpha -\gamma \right ) & =\sin \alpha \cos \gamma -\cos \alpha \sin \gamma \\ \cos \left ( \alpha +\gamma \right ) & =\cos \alpha \cos \gamma -\sin \alpha \sin \gamma \\ \color{green}\cos \left ( \alpha -\gamma \right ) & =\cos \alpha \cos \gamma +\sin \alpha \sin \gamma \\ \tan \left ( \alpha +\gamma \right ) & =\displaystyle \frac{\tan \alpha +\tan \gamma }{1-\tan \alpha \tan \gamma } \\ \color{green}\tan \left ( \alpha -\gamma \right ) & =\displaystyle \frac{\tan \alpha -\tan \gamma }{1+\tan \alpha \tan \gamma } \end{cases}.

Untuk bentuk \color{blue}\sin \left ( \alpha +\beta \right )=\sin \alpha \cos \beta +\cos \alpha \sin \beta akan diberikan buktinya sebagaimana berikut ini
Perhatikanlah ilustrasi segitiga ABC berikut

Sebagai buktinya adalah, untuk ilustrasi segitiga kita tambah keterangan seabagimana berikut, yaitu

Perhatikanlah ΔAA’C dan ΔAA’B

\color{magenta}\begin{aligned}\displaystyle \frac{AC}{\sin 90^{0}}&=\displaystyle \frac{CA'}{\sin \alpha }=\displaystyle \frac{AA'}{\sin \angle C}\\ AA'&=AC.\sin \angle C\\ &=AC.\sin \left ( 90^{0}-\alpha \right )\\ &=AC.\cos \alpha\\ &\color{blue}\textnormal{dengan cara yang kurang lebih sama akan diperoleh juga}\\ AA'&=AB.\cos \beta\\ &\textnormal{selanjutnya kita tentukan luas seperti perinyah soal}\\ \left [ ABC \right ]&=\left [ AA'C \right ]+\left [ AA'B \right ]\\ \displaystyle \frac{1}{2}.AB.AC.\sin \left ( \alpha +\beta \right )&=\displaystyle \frac{1}{2}.AC.AA'.\sin \alpha +\displaystyle \frac{1}{2}.AB.AA'.\sin \beta \\ \sin \left ( \alpha +\beta \right )&=\displaystyle \frac{AC.AA'.\sin \alpha }{AB.AC}+\displaystyle \frac{AB.AA'.\sin \beta }{AB.AC}\\ &=\displaystyle \frac{AA'}{AB}.\sin \alpha +\displaystyle \frac{AA'}{AC}.\sin \beta \\ &=\displaystyle \frac{\left ( AB.\cos \beta \right )}{AB}.\sin \alpha +\displaystyle \frac{\left ( AC.\cos \alpha \right )}{AC}.\sin \beta \\ \color{black}\sin \left ( \alpha +\beta \right )&=\sin \alpha .\cos \beta +\cos \alpha .\sin \beta \quad \color{black}\blacksquare \end{aligned}.

B. Rumus Jumlah dan Selisih Sinus dan Kosinus

\color{blue}\begin{cases} \sin X+\sin Y & =2\sin \frac{1}{2}\left (X+Y \right )\cos \frac{1}{2}\left ( X-Y \right ) \\ \sin X-\sin Y & =2\cos \frac{1}{2}\left (X+Y \right )\sin \frac{1}{2}\left ( X-Y \right )\\ \cos X+\cos Y & =2\cos \frac{1}{2}\left (X+Y \right )\cos \frac{1}{2}\left ( X-Y \right ) \\ \cos X-\cos Y & =-2\sin \frac{1}{2}\left (X+Y \right )\sin \frac{1}{2}\left ( X-Y \right ) \end{cases}.

Untuk mendapatkan rumus di atas, ambil contoh untuk persamaan rumus no,3, yaitu:

\color{blue}\begin{array}{|cll|c|}\hline &\begin{aligned}\cos \left ( \alpha +\gamma \right )&=\cos \alpha \cos \beta -\sin \alpha \sin \gamma \\ \cos \left ( \alpha -\gamma \right )&=\cos \alpha \cos \beta +\sin \alpha \sin \gamma \\ \end{aligned}&&\\\hline &&+&\textrm{Misalkan}\\ &\begin{aligned}\cos \left ( \alpha +\gamma \right )+\cos \left ( \alpha -\gamma \right )&=2\cos \alpha \cos \gamma \end{aligned}&&X=\alpha +\gamma \\ &&&Y=\alpha -\gamma\\ &\color{magenta}\textbf{Proses perubahannya adalah sebagai berikut}&&\\\hline &\underset{\textrm{perhatikan}}{\underbrace{\begin{matrix} \begin{array}{ccc}\\ &X=\alpha +\gamma &\\ &Y=\alpha -\gamma &+\\\hline{2-2}\\ &X+Y=2\alpha &\\ &2\alpha =X+Y&\\ &\alpha =\displaystyle \frac{X+Y}{2}& \end{array} & \textrm{dan} & \begin{array}{ccc}\\ &X=\alpha +\gamma &\\ &Y=\alpha -\gamma &-\\\hline\\ &X-Y=2\gamma &\\ &2\gamma =X-Y&\\ &\gamma =\displaystyle \frac{X-Y}{2}& \end{array} \end{matrix}}} &&\\\hline &\color{magenta}\textbf{Sehingga rumus akan menjadi}&&\\ &\cos X+\cos Y =2\cos \displaystyle \frac{\left ( X+Y \right )}{2}\cos \frac{\left ( X- Y\right )}{2}&&\\\hline \end{array}.

. C. Rumus bentuk Perkalian Trigonometri

\color{blue}\begin{array}{|c|c|}\hline \textrm{Sejenis}&\textrm{Tidak Sejenis}\\\hline \begin{cases} 2\cos \alpha \cos \gamma &=\cos \left ( \alpha +\gamma \right )+\cos \left ( \alpha -\gamma \right )\\ - 2\sin \alpha \sin \gamma &=\cos \left ( \alpha +\gamma \right )-\cos \left ( \alpha -\gamma \right ) \end{cases}&\begin{cases} 2\sin \alpha \cos \gamma &=\sin \left ( \alpha +\gamma \right )+\sin \left ( \alpha -\gamma \right )\\ 2\cos \alpha \sin \gamma &=\sin \left ( \alpha +\gamma \right )-\sin \left ( \alpha -\gamma \right ) \end{cases}\\\hline \end{array}.

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