Contoh Soal 4 Persamaan Trigonometri (Matematika Peminatan Kelas XI)

$\begin{array}{ll}\\ 16.&\textrm{Jika}\: \: \tan^{2} x +\sec x =5 \: \: \textrm{untuk}\\ &0\leq x\leq \displaystyle \frac{\pi }{2}\: \: \textrm{maka nilai} \: \: \cos x=....\\ &\begin{array}{llll}\\ \textrm{a}.&0\\ \color{red}\textrm{b}.&\displaystyle \frac{1}{2}\\ \textrm{c}.&\displaystyle \frac{1}{3}\\ \textrm{d}.&\displaystyle \frac{1}{\sqrt{2}}\\ \textrm{e}.&\displaystyle \frac{1}{2}\sqrt{3} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}\textrm{Ingat bahwa}&\: \: 0\leq x\leq \displaystyle \frac{\pi }{2}\\ \textrm{berarti sudu}&\textrm{t}\: \: x\: \: \textrm{berada di kuadran I}\\ \textrm{sehingga ak}&\textrm{an menyebabkan nilai}\\ & \color{magenta}\cos x=+\\ \color{black}\textrm{Selanjutnya}&\\ \tan^{2} x +\sec x &=5\\ \sec ^{2}x-1+\sec x&=5\\ \sec ^{2}x+\sec x-6&=0\\ (\sec x+3)(\sec x-2)&=0\\ \sec x=-3\: \: \textrm{atau}&\sec x=2\\ \textrm{untuk}\: \: \sec x&=-3\: \: (\color{red}\textrm{tidak memenuhi})\\ \textrm{untuk}\: \: \sec x&=2\: \: (\color{magenta}\textrm{memenuhi})\\ \color{black}\textrm{Selanjutnya}&\: \color{black}\textrm{lagi}\\ \sec x&=2\\ \displaystyle \frac{1}{\cos x}&=2\\ \cos x&=\color{magenta}\displaystyle \frac{1}{2} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 17.&\textrm{Nilai}\\ &\left ( \sin A+\cos A \right )^{2}+\left ( \sin A-\cos A \right )^{2}=....\\ &\begin{array}{llll}\\ \textrm{a}.&1\\ \color{red}\textrm{b}.&\displaystyle 2\\ \textrm{c}.&\displaystyle 3\\ \textrm{d}.&\displaystyle 3\cos A\\ {e}.&\displaystyle 4\sin A \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}&\left ( \sin A+\cos A \right )^{2}+\left ( \sin A-\cos A \right )^{2}\\ &=\sin ^{2}A+2\sin A\cos A+\cos ^{2}A\\ &\quad +\sin ^{2}A-2\sin A\cos A+\cos ^{2}A\\ &=1+1\\ &=2 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 18.&\textrm{Nilai}\: \: \sqrt{\displaystyle \frac{1+\sin A}{1-\sin A}}=....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&\sec A+\tan A\\ \textrm{b}.&\sec ^{2}A+\tan ^{2}A\\ \textrm{c}.&\sec ^{2}A-\tan ^{2}A\\ \textrm{d}.&\tan ^{2}A-\sec ^{2}A\\ {e}.&\sec A\times \tan A \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}&\sqrt{\displaystyle \frac{1+\sin A}{1-\sin A}}\\ &=\sqrt{\displaystyle \frac{1+\sin A}{1-\sin A}\times \frac{1+\sin A}{1+\sin A}}\\ &=\sqrt{\displaystyle \frac{\left ( 1+\sin A \right )^{2}}{1-\sin ^{2}A}}\\ &=\sqrt{\displaystyle \frac{\left ( 1+\sin A \right )^{2}}{\cos ^{2}A}}\\ &=\displaystyle \frac{1+\sin A}{\cos A}\\ &=\displaystyle \frac{1}{\cos A}+\frac{\sin A}{\cos A}\\ &=\sec A+\tan A \end{aligned} \end{array}$

$\begin{array}{ll}\\ 19.&\textrm{Jika}\: \: 0^{\circ}\leq \theta \leqslant 90^{\circ},\: \textrm{maka nilai}\\ &\left ( \displaystyle \frac{5\cos \theta -4}{3-5\sin \theta }-\frac{3+5\sin \theta }{4+5\cos \theta } \right )=....\\ &\begin{array}{llll}\\ \textrm{a}.&-1\\ \color{red}\textrm{b}.&\displaystyle 0\\ \textrm{c}.&\displaystyle \frac{1}{4}\\ \textrm{d}.&\displaystyle \frac{1}{2}\\ {e}.&\displaystyle 1 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}&\left ( \displaystyle \frac{5\cos \theta -4}{3-5\sin \theta }-\frac{3+5\sin \theta }{4+5\cos \theta } \right )\\ &=\left ( \displaystyle \frac{5\cos \theta -4}{3-5\sin \theta }\times \frac{4+5\cos \theta }{4+5\cos \theta } \right )\\ &\qquad-\left ( \displaystyle \frac{3+5\sin \theta }{4+5\cos \theta }\times \frac{3-5\sin \theta }{3-5\sin \theta } \right )\\ &=\displaystyle \frac{25\cos ^{2}-16-\left ( 9-25\sin ^{2}\theta \right )}{(3-5\sin \theta )(4+5\cos \theta )}\\ &=\displaystyle \frac{25\left ( \sin ^{2}\theta +\cos ^{2}\theta \right )-25}{(3-5\sin \theta )(4+5\cos \theta )}\\ &=\displaystyle \frac{0}{(3-5\sin \theta )(4+5\cos \theta )}\\ &=\color{magenta}0 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 20.&\textrm{Nilai}\\ &\left ( 1+\cot \theta -\csc \theta \right )\left ( 1+\tan \theta +\sec \theta \right )=....\\ &\begin{array}{llll}\\ \textrm{a}.&-2\\ \textrm{b}.&\displaystyle -1\\ \textrm{c}.&\displaystyle 0\\ \textrm{d}.&\displaystyle 1\\ \color{red}\textrm{e}.&\displaystyle 2 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{e}\\ &\color{blue}\begin{aligned}&\left ( 1+\cot \theta -\csc \theta \right )\left ( 1+\tan \theta +\sec \theta \right )\\ &=\left ( 1+\displaystyle \frac{\cos \theta }{\sin \theta } -\frac{1}{\sin \theta } \right )\left ( 1+\frac{\sin \theta }{\cos \theta } +\frac{1}{\cos \theta } \right )\\ &=\left ( \displaystyle \frac{\sin \theta +\cos \theta -1}{\sin \theta } \right )\left ( \displaystyle \frac{\cos \theta +\sin \theta +1}{\cos \theta } \right )\\ &=\displaystyle \frac{\left (\sin \theta +\cos \theta \right )^{2}-1}{\sin \theta \cos \theta }\\ &=\displaystyle \frac{1+2\sin \theta \cos \theta -1}{\sin \theta \cos \theta }\\ &=2 \end{aligned} \end{array}$