Contoh Soal Turunan Fungsi Trigonomometri (Bagian 1)

$\begin{array}{l}\\ 1.& \text{Sebuah partikel bergerak menurut}f(t)=3\sin {\displaystyle \frac{1}{6}t}\\ & \text{(dalam cm)}.\text{Posisi partikel saat}t=5\pi \text{detik adalah}\\ & ....\text{cm}\\ & \begin{array}{l}\\ \text{a}.& -{\displaystyle \frac{3}{2}\sqrt{3}}\\ \text{b}.& -{\displaystyle \frac{3}{2}}\\ \text{c}.& {\displaystyle \frac{3}{2}}\\ \text{d}.& {\displaystyle \frac{3}{2}\sqrt{2}}\\ \text{e}.& {\displaystyle \frac{3}{2}\sqrt{3}}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}f(t)& =3\sin {\displaystyle \frac{1}{6}t}\\ \text{posi}& \text{si benda saat}t=5\pi \text{detik adalah}:\\ f(5\pi )& =3\sin {\displaystyle \frac{1}{6}(5\pi )}\\ & =3\sin {\displaystyle \frac{5}{6}\pi }\\ & =3\sin {150}^{\circ }\\ & =3\left({\displaystyle \frac{1}{2}}\right)\\ & ={\displaystyle \frac{3}{2}}\end{array}\end{array}$

$\begin{array}{l}\\ 2.& \text{Sebuah objek bergerak melingkar mengikuti rumus}\\ & f(t)=-2\sin t\text{pada waktu}t.\text{Fungsi kecepatan}\\ & \text{dari benda tersebut adalah}....\text{cm}\\ & \begin{array}{l}\\ \text{a}.& -2\cos 2t\\ \text{b}.& -2\cos t\\ \text{c}.& -\cos 2t\\ \text{d}.& \cos t\\ \text{e}.& 2\cos t\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}f(t)& =-2\sin t\\ \text{fun}& \text{gsi kecepatannya adalah}:\\ v(t)& =f^{\prime }(t)=-2\cos t\end{array}\end{array}$

$\begin{array}{l}\\ 3.& \text{Kedudukan sebuah partikel yang bergerak}\\ & \text{mengikuti model}f(t)=\cos 5t\text{(dalam cm)}\\ & \text{pada saat}t=\pi \text{detik adalah}....\\ & \begin{array}{l}\\ \text{a}.& \text{1 cm di bawah sumbu horizontal}\\ \text{b}.& \text{1 cm di atas sumbu horizontal}\\ \text{c}.& \text{5 cm di bawah sumbu horizontal}\\ \text{d}.& \text{5 cm di atas sumbu horizontal}\\ \text{e}.& \text{tepat pada sumbu horizontal}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}f(t)& =\cos 5t\\ \text{kedu}& \text{dukannya adalah}:\\ f\left(\pi \right)& =\cos 5(\pi )\\ & =\cos \pi \\ & =-1\end{array}\end{array}$

$\begin{array}{l}\\ 4.& \text{Kedudukan sebuah partikel yang bergerak}\\ & \text{mengikuti rumus}f(t)=2\sin t+\cos {\displaystyle \frac{1}{2}t}\\ & \text{Selang waktu berikut yang menunjukkan}\\ & \text{posisi partikel di bawah sumbu X dan}\\ & \text{bergerak naik adalah}....\\ & \begin{array}{l}\\ \text{a}.& 0<t<{\displaystyle \frac{1}{2}\pi }\\ \text{b}.& {\displaystyle \frac{1}{2}\pi <t<\pi }\\ \text{c}.& {\displaystyle \frac{1}{2}\pi <t<{\displaystyle \frac{3}{2}\pi }}\\ \text{d}.& \pi <t<{\displaystyle \frac{3}{2}\pi }\\ \text{e}.& {\displaystyle \frac{3}{2}\pi <t<2\pi }\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{d}}\\ & \begin{array}{rl}f(t)& =2\sin t+\cos {\displaystyle \frac{1}{2}t}\\ \text{kedu}& \text{dukannya adalah}:\end{array}\end{array}$