Contoh Soal 6 Persamaan Trigonometri (Matematika Peminatan Kelas XI)

$\begin{array}{l}\\ 26.& \text{Jika nilai}\cot A+\cos A=x\\ & \text{dan}\cot A-\cos A=y,\\ & \text{maka nilai}(x^2-y^2)=....\\ & \begin{array}{l}\\ \text{a}.& 4\sqrt{xy}\\ \text{b}.& 2\sqrt{xy}\\ \text{c}.& xy\\ \text{d}.& 2\\ \text{e}.& 4\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}xy& =(\cot A+\cos A)(\cot A-\cos A)\\ & ={\cot }^{2}A-{\cos }^{2}A\\ & ={\displaystyle \frac{{\cos }^{2}A}{{\sin }^{2}A}-{\cos }^{2}A}\\ & ={\displaystyle \frac{{\cos }^{2}A}{{\sin }^{2}A}-{\cos }^{2}A\times \frac{{\sin }^{2}A}{{\sin }^{2}A}}\\ & ={\displaystyle \frac{{\cos }^{2}A}{{\sin }^{2}A}(1-{\sin }^{2}A)}\\ & ={\displaystyle \frac{{\cos }^{4}A}{{\sin }^{2}A}}\\ \sqrt{xy}& ={\displaystyle \frac{\cos A}{\sin A}\times \cos A}\\ \text{Se}& \text{lanjutnya}\\ x^2-y^2& ={(\cot A+\cos A)}^2-{(\cot A-\cos A)}^2\\ (x+y)& (x-y)=(\cot A+\cos A+\cot A-\cos A)\\ & \times (\cot A+\cos A-(\cot A-\cos A))\\ x^2-y^2& =2\cot A\times 2\cos A\\ & =4\times {\displaystyle \frac{\cos A}{\sin A}\times \cos A}\\ & =4\sqrt{xy}\end{array}\end{array}$

$\begin{array}{l}\\ 27.& \text{Jika nilai}\cos A+\sin A=\sqrt{2}\cos A\\ & \text{maka nilai}(\cos A-\sin A)=....\\ & \begin{array}{l}\\ \text{a}.& -\sqrt{2}\cos A\\ \text{b}.& -\sqrt{2}\sin A\\ \text{c}.& \sqrt{2}\sin A\\ \text{d}.& {\displaystyle \frac{1}{\sqrt{2}}\sec A}\\ \text{e}.& {\displaystyle \frac{1}{\sqrt{2}}\csc A}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}\cos A+\sin A& =\sqrt{2}\cos A\\ {(\cos A+\sin A)}^2& ={(\sqrt{2}\cos A)}^2\\ 1+2\sin A\cos A& =2{\cos }^{2}A\\ 2\sin A\cos A& =2{\cos }^{2}A-1\\ \text{maka}& \\ {(\cos A-\sin A)}^2& =1-2\sin A\cos A\\ & =1-(2{\cos }^{2}A-1)\\ & =2-2{\cos }^{2}A\\ & =2(1-{\cos }^{2}A)\\ & =2{\sin }^{2}A\\ \cos A-\sin A& =\sqrt{2{\sin }^{2}A}\\ & =\sin A\sqrt{2}\\ & =\sqrt{2}.\sin A\end{array}\end{array}$

DAFTAR PUSTAKA

1. Sembiring, S., Zulkifli, M., Marsito, dan Rusdi, I. 2017. Matematika untuk Siswa SMA/MA Kelas XI Kelompok Peminatan Matematika dan Ilmu-Ilmu Alam. Bandung: SEWU.
2. Sukino. 2016. Matematika untuk SMA/MA Kelas XI Kelompok Peminatan Matematika dan Ilmu-Ilmu Alam. Jakarta: ERLANGGA