$\begin{array}{ll}\\ 51.&\textrm{Nilai dari}\: \: \displaystyle \frac{1}{\sec ^{2}A}+\frac{1}{\csc ^{2}A}=....\\ &\begin{array}{llll}\\ \textrm{a}.&-\infty \\ \textrm{b}.&\displaystyle -1\\ \textrm{c}.&\displaystyle 0\\ \color{red}\textrm{d}.&\displaystyle 1\\ {e}.&\displaystyle \infty \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\begin{aligned}&\displaystyle \frac{1}{\sec ^{2}A}+\frac{1}{\csc ^{2}A}\\ &=\cos ^{2}A+\sin ^{2}=\color{red}1 \end{aligned} \end{array}$
$\begin{array}{ll}\\ 52.&\textrm{Nilai dari}\: \: \displaystyle \frac{\tan B+\tan C}{\cot B+\cot C}=....\\ &\begin{array}{llll}\\ \textrm{a}.&\cot B\times \cot C \\ \color{red}\textrm{b}.&\displaystyle \tan B\times \tan C\\ \textrm{c}.&\displaystyle \sec B\times \csc C\\ \textrm{d}.&\displaystyle \tan B\times \cot C\\ {e}.&\displaystyle \tan B\times \csc C \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\begin{aligned}&\displaystyle \frac{\tan B+\tan C}{\cot B+\cot C}\\ &=\displaystyle \frac{\tan B+\tan C}{\displaystyle \frac{1}{\tan B}+\frac{1}{\tan C}}\\ &=\displaystyle \frac{\tan B+\tan C}{\left ( \displaystyle \frac{\tan B+\tan C}{\tan B\times \tan C} \right )}\\ &=\color{red}\tan B\times \tan C \end{aligned} \end{array}$
$\begin{array}{ll}\\ 53.&\textrm{Nilai dari}\\ &\displaystyle \frac{\tan A}{\sec A-1}+\frac{\tan A}{\sec A+1}=....\\ &\begin{array}{llll}\\ \textrm{a}.&2\tan A \\ \textrm{b}.&2\cot A\\ \textrm{c}.&\displaystyle 2\sec A\\ \color{red}\textrm{d}.&\displaystyle 2\csc A\\ {e}.&\displaystyle 2\tan A.\sec A \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\begin{aligned}&\displaystyle \frac{\tan A}{\sec A-1}+\frac{\tan A}{\sec A+1}\\ &=\tan A\left (\displaystyle \frac{1}{\displaystyle \frac{1}{\cos A}-1}+\frac{1}{\displaystyle \frac{1}{\cos A}+1} \right )\\ &=\displaystyle \frac{\sin A}{\cos A}\left ( \displaystyle \frac{\cos A}{1-\cos A}+\frac{\cos A}{1+\cos A} \right )\\ &=\displaystyle \frac{\sin A}{1-\cos A}+\frac{\sin A}{1+\cos A}\\ &=\displaystyle \frac{\sin A(1+\cos A)+\sin A(1-\cos A)}{(1-\cos A)(1+\cos A)}\\ &=\displaystyle \frac{2\sin A}{1-\cos ^{2}}\\ &=\displaystyle \frac{2\sin A}{\sin ^{2}A}\\ &=\displaystyle \frac{2}{\sin A}\\ &=\color{red}2\csc A \end{aligned}\\\\ &\textrm{Sebagai catatanya}\\ &\textrm{Anda bisa gunakan cara yang lain} \end{array}$
$\begin{array}{ll}\\ 54.&\textrm{Nilai}\: \: x\: \: \textrm{yang memenuhi persamaan}\\ &\sin \left ( 2x-20^{\circ} \right )=-\cos \left ( 3x+50^{\circ} \right )\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&-30^{\circ}\\ \textrm{b}.&-25^{\circ}\\ \color{red}\textrm{c}.&20^{\circ}\\ \textrm{d}.&25^{\circ}\\ {e}.&30^{\circ} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\begin{aligned}\sin \left ( 2x-20^{\circ} \right )&=-\cos \left ( 3x+50^{\circ} \right )\\ \sin \left ( 20^{\circ}-2x \right )&=\cos \left ( 3x+50^{\circ} \right )\\ \sin A&=\cos B,\: \: \color{black}\textrm{artinya}\\ A+B&=90^{\circ},\: \: \color{magenta}\textrm{maka}\\ \left ( 20^{\circ}-2x \right )+\left ( 3x+50^{\circ} \right )&=90^{\circ}\\ x+70^{\circ}&=90^{\circ}\\ x&=90^{\circ}-70^{\circ}\\ &=\color{red}20^{\circ} \end{aligned} \end{array}$
$\begin{array}{ll}\\ 55.&\textrm{Nilai}\: \: x\: \: \textrm{yang memenuhi persamaan}\\ &\tan \left ( 2x+60^{\circ} \right )=\cot \left ( 90^{\circ}-3x \right )\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&20^{\circ}\\ \textrm{b}.&30^{\circ}\\ \textrm{c}.&40^{\circ}\\ \textrm{d}.&50^{\circ}\\ \color{red}\textrm{e}.&60^{\circ} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{e}\\ &\begin{aligned}\tan \left ( 2x+60^{\circ} \right )&=\cot \left ( 90^{\circ}-3x \right )\\ \tan (2x+60^{\circ})&=\tan 3x\\ 2x+60^{\circ}&=3x\\ 2x-3x&=-60^{\circ}\\ -x&=-60^{\circ}\\ x&=\color{red}60^{\circ} \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 56.&\textrm{Jika nilai}\: \: \cot A+\cos A=x\\ &\textrm{dan}\: \: \cot A-\cos A=y,\\ &\textrm{maka nilai}\: \: \left ( x^{2}-y^{2} \right )=\: ....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&4\sqrt{xy}\\ \textrm{b}.&2\sqrt{xy}\\ \textrm{c}.&xy\\ \textrm{d}.&2\\ \textrm{e}.&4 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\begin{aligned}xy&=\left (\cot A+\cos A \right )\left ( \cot A-\cos A \right )\\ &=\cot ^{2}A-\cos ^{2}A\\ &=\displaystyle \frac{\cos^{2}A }{\sin ^{2}A}-\cos ^{2}A\\ &=\displaystyle \frac{\cos^{2}A }{\sin ^{2}A}-\cos ^{2}A\times \frac{\sin ^{2}A}{\sin ^{2}A}\\ &=\displaystyle \frac{\cos ^{2}A}{\sin ^{2}A}\left ( 1-\sin ^{2}A \right )\\ &=\displaystyle \frac{\cos ^{4}A}{\sin ^{2}A}\\ \sqrt{xy}&=\displaystyle \frac{\cos A}{\sin A}\times \cos A\\ \color{black}\textrm{Se}&\color{black}\textrm{lanjutnya}\\ x^{2}-y^{2}&=\left (\cot A+\cos A \right )^{2}-\left ( \cot A-\cos A \right )^{2}\\ (x+y)&(x-y)=(\cot A+\cos A+\cot A-\cos A)\\ &\qquad\times (\cot A+\cos A-(\cot A-\cos A))\\ x^{2}-y^{2}&=2\cot A\times 2\cos A\\ &=4\times \displaystyle \frac{\cos A}{\sin A}\times \cos A\\ &=\color{red}4\sqrt{xy} \end{aligned} \end{array}$
$\begin{array}{ll}\\ 57.&\textrm{Jika nilai}\: \: \cos A+\sin A=\sqrt{2}\cos A\\ &\textrm{maka nilai}\: \: \left ( \cos A-\sin A \right )=\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&-\sqrt{2}\cos A\\ \textrm{b}.&-\sqrt{2}\sin A\\ \color{red}\textrm{c}.&\sqrt{2}\sin A\\ \textrm{d}.&\displaystyle \frac{1}{\sqrt{2}}\sec A\\ \textrm{e}.&\displaystyle \frac{1}{\sqrt{2}}\csc A \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\begin{aligned}\cos A+\sin A&=\sqrt{2}\cos A\\ \left (\cos A+\sin A \right )^{2}&=\left (\sqrt{2}\cos A \right )^{2}\\ 1+2\sin A\cos A&=2\cos ^{2}A\\ 2\sin A\cos A&=2\cos ^{2}A-1\\ \textrm{maka}&\\ \left (\cos A-\sin A \right )^{2}&=1-2\sin A\cos A\\ &=1-\left ( 2\cos ^{2}A-1 \right )\\ &=2-2\cos ^{2}A\\ &=2\left ( 1-\cos ^{2}A \right )\\ &=2\sin ^{2}A\\ \cos A-\sin A&=\sqrt{2\sin ^{2}A}\\ &=\sin A\sqrt{2}\\ &=\color{red}\sqrt{2}.\sin A \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 58.&\textrm{Nilai}\: \: \cos \gamma \left ( \csc \gamma +\tan \gamma \right ) \\ & \textrm{adalah ekivalen dengan}\: ....\\ &\begin{array}{llllllll}\\ \textrm{a}.&\color{red}\cot \gamma +\sin \gamma \\ \textrm{b}.&\tan \gamma +\cos \gamma \\ \textrm{c}.&\cot \gamma -\sin \gamma \\ \textrm{d}.&\tan \gamma -\cos \gamma \\ \textrm{e}.&\cot \gamma -\tan \gamma \end{array}\\\\ &\textbf{Jawab}:\quad \color{red}\textbf{a}\\ &\begin{aligned}\cos \gamma \left ( \csc \gamma +\tan \gamma \right )&=\cos \gamma \left ( \displaystyle \frac{1}{\sin \gamma } +\displaystyle \frac{\sin \gamma }{\cos \gamma } \right )\\ &=\cos \gamma \left ( \displaystyle \frac{\cos \gamma +\sin ^{2}\gamma }{\sin \gamma \cos \gamma } \right )\\ &=\displaystyle \frac{\cos \gamma +\sin ^{2}\gamma }{\sin \gamma }\\ &=\displaystyle \frac{\cos \gamma }{\sin \gamma }+\displaystyle \frac{\sin ^{2}\gamma }{\sin \gamma }\\ &=\color{red}\cot \gamma +\sin \gamma \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 59.&\textrm{Jika diketahui}\: \: \sin \theta \cos \theta =\displaystyle \frac{3}{8},\\ &\textrm{maka nilai}\: \: \displaystyle \frac{1}{\sin \theta }-\displaystyle \frac{1}{\cos \theta }\: \: \textrm{adalah}=....\\ &\begin{array}{llllllll}\\ \textrm{a}.&\displaystyle \frac{1}{4}\\ \textrm{b}.&\displaystyle \frac{1}{2}\\ \textrm{c}.&\displaystyle \frac{3}{4} \\ \textrm{d}.&\color{red}\displaystyle \frac{4}{3}\\ \textrm{e}.&\displaystyle 4 \end{array}\\\\ &\textbf{Jawab}:\quad \color{red}\textbf{d}\\ &\begin{aligned}&\displaystyle \frac{1}{\sin \theta }-\displaystyle \frac{1}{\cos \theta }\\ &=\sqrt{\left ( \displaystyle \frac{1}{\sin \theta }-\displaystyle \frac{1}{\cos \theta } \right )^{2}}\\ &=\sqrt{\displaystyle \frac{1}{\sin ^{2}\theta }-\displaystyle \frac{2}{\sin \theta \cos \theta }+\displaystyle \frac{1}{\cos ^{2}\theta }}\\ &=\sqrt{\displaystyle \frac{\cos ^{2}\theta +\sin ^{2}\theta }{\sin ^{2}\theta \cos ^{2}\theta }-\displaystyle \frac{2}{\sin \theta \cos \theta }}\\ &=\sqrt{\displaystyle \frac{1}{\left ( \displaystyle \frac{3}{8} \right )^{2}}-\displaystyle \frac{2}{\left ( \displaystyle \frac{3}{8} \right )}}=\sqrt{\displaystyle \frac{1}{\displaystyle \frac{9}{64}}-\displaystyle \frac{2}{\displaystyle \frac{3}{8}}}=\sqrt{\displaystyle \frac{64}{9}-\displaystyle \frac{16}{3}}\\ &=\sqrt{\displaystyle \frac{64}{9}-\displaystyle \frac{48}{9}}=\sqrt{\displaystyle \frac{16}{9}}=\color{red}\displaystyle \frac{4}{3} \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 60.&\textrm{Bentuk}\: \: \displaystyle \frac{\left ( \sin \alpha +\sin \beta \right )\left ( \sin \alpha -\sin \beta \right )}{\left ( \cos \alpha \cos \beta \right )^{2}}\\ & \textrm{ekivalen dengan}\: ....\\ &\begin{array}{llllllll}\\ \textrm{a}.&\tan \alpha -\tan \beta \\ \textrm{b}.&\tan \alpha +\tan \beta \\ \textrm{c}.&\color{red}\tan ^{2}\alpha -\tan ^{2}\beta \\ \textrm{d}.&\tan ^{2}\alpha +\tan ^{2}\beta \\ \textrm{e}.&\tan ^{3}\alpha -\tan ^{3}\beta \end{array}\\\\ &\textbf{Jawab}:\quad \color{red}\textbf{c}\\ &\begin{aligned}&\displaystyle \frac{\left ( \sin \alpha +\sin \beta \right )\left ( \sin \alpha -\sin \beta \right )}{\left ( \cos \alpha \cos \beta \right )^{2}}\\ &=\displaystyle \frac{\sin ^{2}\alpha -\sin ^{2}\beta }{\left ( \cos \alpha \cos \beta \right )^{2}}\\ &=\displaystyle \frac{\sin ^{2}\alpha -\sin ^{2}\alpha \sin ^{2}\beta -\sin ^{2}\beta +\sin ^{2}\alpha \sin ^{2}\beta}{\left ( \cos \alpha \cos \beta \right )^{2}}\\ &=\displaystyle \frac{\sin ^{2}\alpha\left ( 1-\sin ^{2}\beta \right ) -\sin ^{2}\beta \left ( 1-\sin ^{2}\alpha \right )}{\left ( \cos \alpha \cos \beta \right )^{2}}\\ &=\displaystyle \frac{\sin ^{2}\alpha \cos ^{2}\beta -\sin ^{2}\beta \cos ^{2}\alpha }{\left ( \cos \alpha \cos \beta \right )^{2}}\\ &=\displaystyle \frac{\sin ^{2}\alpha \cos ^{2}\beta }{\left ( \cos \alpha \cos \beta \right )^{2}}-\displaystyle \frac{\sin ^{2}\beta \cos ^{2}\alpha }{\left ( \cos \alpha \cos \beta \right )^{2}}\\ &=\displaystyle \frac{\sin ^{2}\alpha }{\cos ^{2}\alpha }-\displaystyle \frac{\sin ^{2}\beta }{\cos ^{2}\beta }\\ &=\color{red}\tan ^{2}\alpha -\tan ^{2}\beta \end{aligned} \end{array}$.
DAFTAR PUSTAKA
- 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.
- Sukino. 2016. Matematika untuk SMA/MA Kelas XI Kelompok Peminatan Matematika dan Ilmu-Ilmu Alam. Jakarta: ERLANGGA
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