Soal Persiapan Penilaian Tengah Semester Gasal Kelas XI Pendalaman Matematika (Peminatan)

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

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

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

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

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

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

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