Latihan Soal 1 Persiapan PAS Gasal Matematika Peminatan Kelas XI (Persamaan dan Rumus Jumlah dan Selisih Trigonometri)

 $\begin{array}{ll}\\ 1.&\textrm{Nilai}\: \: 75^{\circ}\: \: \textrm{jika dinyatakan ke radian}\\ &\textrm{adalah}\: \: ....\: \: \textrm{radian}\\\\ &\textrm{a}.\quad \displaystyle \frac{1}{3}\pi \\\\ &\textrm{b}.\quad \displaystyle \frac{5}{6}\pi \\\\ &\textrm{c}.\quad \displaystyle \color{red}\frac{5}{12}\pi \\\\ &\textrm{d}.\quad \displaystyle \frac{7}{12}\pi \\\\ &\textrm{e}.\quad \displaystyle \frac{9}{12}\pi \\\\ &\textbf{Jawab}:\\ &\begin{aligned}\textrm{Diketah}&\textrm{ui bahwa}\\ 180^{\circ}&=\pi \: \: \: radian\\ 1^{\circ}&=\displaystyle \frac{\pi }{180}\: \: \: radian\\ 75\times 1^{\circ}&=75\times \displaystyle \frac{\pi }{180}\: \: \: radian\\ 75^{\circ}&=\displaystyle \frac{5}{12}\pi \: \: \: radian \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 2.&\textrm{Jika}\: \: \tan \theta =\displaystyle \frac{5}{12}\: \: \textrm{untuk}\: \: 0^{\circ}\leq \theta \leq 90^{\circ}\\ &\textrm{maka}\: \: \cos \theta \: \: \textrm{adalah}\: ....\\\\ &\textrm{a}.\quad \displaystyle \frac{5}{13} \\\\ &\textrm{b}.\quad \displaystyle \color{red}\frac{12}{13} \\\\ &\textrm{c}.\quad \displaystyle \frac{13}{5} \\\\ &\textrm{d}.\quad \displaystyle \frac{13}{12} \\\\ &\textrm{e}.\quad \displaystyle \frac{12}{5} \\\\ &\textbf{Jawab}:\\ &\textrm{Perhatikanlah gambar segitiga berikut} \end{array}$.

$. \qquad\begin{aligned}\textrm{Diketah}&\textrm{ui bahwa}\\ \tan \theta &=\displaystyle \frac{5}{12}\: ,\: \textrm{untuk}\: 0^{\circ}\leq \theta \leq 90^{\circ}\\ &\color{red}\textrm{lihat gambar di atas}\\ &\textrm{dengan dalil Pythagoras akan}\\ &\textrm{didapatkan sisimiringnya}=13\\ \textrm{jadi}&,\: \textrm{nilai dari}\\ \cos \theta &=\displaystyle \frac{12}{13} \end{aligned}$.

$\begin{array}{ll}\\ 3.&\textrm{Perhatikanlah gambar berikut}\\ \end{array}$.

$.\qquad\begin{array}{ll}\\ &\textrm{Panjang BC adalah}\: ....\\ &\textrm{a}.\quad \displaystyle 20\sin 36^{\circ} \\ &\textrm{b}.\quad \displaystyle 20\cos 36^{\circ} \\ &\textrm{c}.\quad \color{red}\displaystyle 20\tan 36^{\circ} \\ &\textrm{d}.\quad \displaystyle 15\\ &\textrm{e}.\quad \displaystyle 16 \\\\ &\textbf{Jawab}:\\ &\begin{aligned}\textrm{Diketah}&\textrm{ui bahwa}\\ \tan 36^{\circ} &=\displaystyle \frac{BC}{20}\\ \Leftrightarrow &\: \color{red}BC\color{black}=20\tan 36^{\circ} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 4.&\textrm{Nilai}\: \: \tan 300^{\circ} \: \: \: \textrm{adalah}\: \: ....\\\\ &\textrm{a}.\quad \displaystyle \sqrt{3} \\\\ &\textrm{b}.\quad \displaystyle \frac{1}{3}\sqrt{3} \\\\ &\textrm{c}.\quad \displaystyle -\frac{1}{3}\sqrt{3} \\\\ &\textrm{d}.\quad \displaystyle \frac{1}{2}\sqrt{3} \\\\ &\textrm{e}.\quad \displaystyle \color{red}-\sqrt{3} \\\\ &\textbf{Jawab}:\\ &\begin{aligned}\tan 300^{\circ}&=\tan \left ( 360^{\circ}-60^{\circ} \right )\\ &=-\tan 60^{\circ}\\ &=\color{red}-\sqrt{3}\\ \textbf{catatan}&: \textrm{ingat sudut berelasi} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 5.&\textrm{Nilai}\: \: \tan 60^{\circ}-\sin 120^{\circ}-\tan 210^{\circ} \: \: \: \textrm{adalah}\: \: ....\\\\ &\textrm{a}.\quad \displaystyle \frac{1}{6}\sqrt{6} \\\\ &\textrm{b}.\quad \displaystyle \frac{1}{3}\sqrt{6} \\\\ &\textrm{c}.\quad \displaystyle -\frac{1}{2}\sqrt{6} \\\\ &\textrm{d}.\quad \displaystyle \frac{1}{3}\sqrt{3} \\\\ &\textrm{e}.\quad \displaystyle \color{red}\frac{1}{6}\sqrt{3} \\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\tan 60^{\circ}-\sin 120^{\circ}-\tan 210^{\circ}\\ &=\tan 60^{\circ}-\sin \left ( 180^{\circ}-60^{\circ} \right )-\tan \left ( 180^{\circ}+30^{\circ} \right )\\ &=\tan 60^{\circ}-\sin 60^{\circ}-\tan 30^{\circ}\\ &=\sqrt{3}-\displaystyle \frac{1}{2}\sqrt{3}-\displaystyle \frac{1}{3}\sqrt{3}\\ &=\left (1-\displaystyle \frac{1}{2}-\frac{1}{3} \right )\sqrt{3}\\ &=\displaystyle \color{red}\frac{1}{6}\sqrt{3} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 6.&\textrm{Nilai}\: \: x \: \: \textrm{positif terkecil yang memenuhi}\\ &\sin x=-\displaystyle \frac{1}{2}\sqrt{3}\: \: \textrm{adalah}\: ....\\\\ &\textrm{a}.\quad \displaystyle 30^{\circ} \\\\ &\textrm{b}.\quad \displaystyle 60^{\circ} \\\\ &\textrm{c}.\quad \displaystyle 120^{\circ} \\\\ &\textrm{d}.\quad \color{red}\displaystyle 240^{\circ} \\\\ &\textrm{e}.\quad \displaystyle 300^{\circ} \\\\ &\textbf{Jawab}:\\ &\begin{aligned}\sin x&=-\frac{1}{2}\sqrt{3}\\ \textrm{Gun}&\textrm{akan rumus persamaan}\\ &\textrm{sederhana, yaitu}:\\ \sin x&=-\sin 60^{\circ}\\ &=\sin \left ( 180^{\circ}+60^{\circ} \right )\\ &=\sin 240^{\circ}\\ x&=\color{red}240^{\circ} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 7.&\textrm{Jika}\: \: \cos x=\displaystyle \frac{2\sqrt{5}}{5} \: \: \textrm{maka nilai}\\ &\cot x\left ( \displaystyle \frac{\pi }{2}-x \right )\: \: \textrm{adalah}\: ....\\\\ &\textrm{a}.\quad \color{red}\displaystyle \frac{1}{2} \\\\ &\textrm{b}.\quad \displaystyle \frac{1}{3} \\\\ &\textrm{c}.\quad \displaystyle \frac{1}{6} \\\\ &\textrm{d}.\quad \displaystyle \frac{1}{7} \\\\ &\textrm{e}.\quad \displaystyle \frac{1}{8} \\\\ &\textbf{Jawab}:\\ &\begin{aligned}\cos x&=\frac{2\sqrt{5}}{5},\: \: \textrm{maka}\\ \sin^{2} x&+\cos ^{2}x=1\\ \sin x&=1-\cos ^{2}x\\ &=\sqrt{1-\cos ^{2}x}=\sqrt{1-\left (\displaystyle \frac{2\sqrt{5}}{5} \right )^{2}}\\ &=\sqrt{1-\displaystyle \frac{20}{25}}=\sqrt{\displaystyle \frac{5}{25}}=\displaystyle \frac{\sqrt{5}}{5}\\ \cot &\left ( \displaystyle \frac{\pi }{2}-x \right )=\tan x,\: \: \textrm{maka}\\ \tan x&=\displaystyle \frac{\sin x}{\cos x}\\ &=\displaystyle \frac{\sqrt{5}}{2\sqrt{5}}\\ &=\color{red}\displaystyle \frac{1}{2} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 8.&\textrm{Pada setiap}\: \: \alpha \: \: \textrm{berlaku}\\ &\tan \alpha +\cos+\tan (-\alpha ) +\cos (-\alpha )=....\\ &\begin{array}{llllll}\\ \textrm{a}.&0\\ \textrm{b}.&2\tan \alpha \\ \textrm{c}.&\color{red}2\cos \alpha \\ \textrm{d}.&2\left ( \tan \alpha +\cos \alpha \right )\\ \textrm{e}.&2\\\\ &&\color{blue}\textbf{SAT Subjeck Test} \end{array}\\ &\\ &\textbf{Jawab}:\quad\color{red}\textbf{c}\\ &\begin{aligned}\tan \alpha &+\cos+\tan (-\alpha ) +\cos (-\alpha )\\ &=\tan \alpha +\cos-\tan \alpha +\cos \alpha \\ &=\color{red}2\cos \alpha \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 9.&\textrm{Jika}\: \: \sin x=-\sin 35^{\circ}\: \: \textrm{untuk}\: \: 90^{\circ}\leq x\leq 270^{\circ}\\ &\textrm{maka nilai}\: \: x\: \: \textrm{adalah}....\\ &\begin{array}{llllllll}\\ \textrm{a}.&\color{red}215^{\circ}\\ \textrm{b}.&235^{\circ}\\ \textrm{c}.&240^{\circ}\\ \textrm{d}.&255^{\circ}\\ \textrm{e}.&270^{\circ} \end{array}\\\\ &\textbf{Jawab}:\quad \color{red}\textbf{a}\\ &\begin{aligned}&\textrm{Diketahui}\: \textrm{bahwa}\: \: \sin x=-\sin 35^{\circ}\\ & \textrm{untuk}\: \: 90^{\circ}\leq x\leq 270^{\circ},\: \textrm{dengan}\\ &\color{blue}\sin x=-\sin 35^{\circ}\\ &\qquad \textrm{(hanya terjadi dikuadran III dan IV)}\\ &\textrm{karena ba}\textrm{tasnya}\: \textrm{hanya untuk kuadran III saja, }\\ &\textrm{maka}\\ &=\sin \left ( 180^{\circ}+35^{\circ} \right )\\ &=\color{red}\sin 215^{\circ} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 10.&\textrm{Jika}\: \: \tan y=\tan 83^{\circ}\: \: \textrm{untuk}\: \: 90^{\circ}< y< 270^{\circ}\\ &\textrm{maka nilai}\: \: x\: \: \textrm{adalah}....\\ &\begin{array}{llllllll}\\ \textrm{a}.&173^{\circ}\\ \textrm{b}.&187^{\circ}\\ \textrm{c}.&\color{red}263^{\circ}\\ \textrm{d}.&268^{\circ}\\ \textrm{e}.&293^{\circ} \end{array}\\\\ &\textbf{Jawab}:\quad \color{red}\textbf{c}\\ &\begin{aligned}\textrm{Diketahui}&\: \textrm{bahwa nilai}\\ \tan y&=\tan 83^{\circ}\: \: \textrm{untuk}\: \: 90^{\circ}< y< 270^{\circ}\\ \tan y&=\tan \left ( 180^{\circ}+83^{\circ} \right ),\\ & \textrm{karena berada dikuadran III}\\ &=\color{red}\tan 263^{\circ} \end{aligned} \end{array}$.