Trigonometri

 A. Ukuran Sudut

Sudut itu sendiri adalah suatu bangun yang dibatasi oleh dua buah sinar atau garis yang berpotongan di sekitar titik potongnya. Sebagai ilustrasinya perhatikanlah gambar berikut

Ada dua buah sinar yaitu  $\overrightarrow{OA}$  dan  $\overrightarrow{OB}$ yang bertemu atau berpotongan di titik $O$ den terbentuklah  $\mathrm{\angle }AOB$.

Sedangkan berkaitan dengan pengukuran sudut nantinya minimal kita mengenal 3 jenis, yaitu: derajat, radian, dan gone atau grade. Berikut hubungan ketiga jenis ikuran sudut yang dimaksud beserta lambang/notasi penulisannya. Untuk anak ditingkat MA/SMA ukuran sudut yang dikenalkan biasanya derajat dan radian, akan tetapi untuk anak SMK ada satu lagi yaitu satuan gone atau grade.

$\begin{array}{|c|}\hline 1\text{keliling}◯={360}^0=2\pi \text{radian}={400}^g\\ \text{atau}\\ {\displaystyle \frac{1}{2}\text{keliling}◯={180}^0=\pi \text{radian}={200}^9}\\ \text{boleh juga}\\ {\displaystyle \frac{1}{4}\text{keliling}◯={90}^0={\displaystyle \frac{1}{2}\pi \text{radian}={100}^9}}\\ \text{Jika ingin diperkecil lagi}\\ {\displaystyle \frac{1}{8}\text{keliling}◯={45}^0={\displaystyle \frac{1}{4}\pi \text{radian}={50}^9}}\\ \hline\end{array}$.

Perhatikan ilustrasi berikut

Sebagai catatan, terkadang sudut dalam satuan derajat dipresentasikan dalam bentuk seksagesimal, yaitu: diubah dalam satuan menit dan detik.

$\begin{array}{|l|}\hline \bullet {\displaystyle 1^{\circ }={60}^{\prime }={3600}^{″}}\\ \bullet 1^{\prime }={60}^{″}\\ \hline\end{array}$.

$\text{CONTOH SOAL}$.

$\begin{array}{l}\\ 1.& \text{Ubahlah sudut berikut ke satuan}\\ & \text{yang diminta}\\ & \text{a}.1^{\circ }=....\text{rad}\\ & \text{b}.1\text{rad}=....{}^0\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}& \begin{array}{rl}\text{a}.{360}^0& =2\pi \text{rad}\\ 360\times 1^0& =2\pi \text{rad}\\ 1^0& ={\displaystyle \frac{2\pi }{360}}\\ & ={\displaystyle \frac{\pi }{180}\text{rad}}\end{array}\\ & \begin{array}{rl}\text{b}.2\pi \text{rad}& ={360}^0\\ 2\pi \times 1\text{rad}& ={360}^0\\ 1\text{rad}& ={\left({\displaystyle \frac{360}{2\pi }}\right)}^0\\ & ={\left({\displaystyle \frac{180}{\pi }}\right)}^0\end{array}\end{array}\end{array}$.

$\begin{array}{l}\\ 2.& \text{Ubahlah sudut berikut ke satuan }\\ & \text{yang diminta}\\ & \text{a}.53,{24}^{\circ }=....\text{(dalam sexagesimal)}\\ & \text{b}.{23}^{\circ }{12}^{\prime }{24}^{″}=....{}^0\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}\text{a}.53,{24}^0& ={53}^0+0,{24}^0\\ & ={53}^0+0,24\times 1^0\\ & ={53}^0+0,24\times {60}^{\prime }\\ & ={53}^0+{14,4}^{\prime }\\ & ={53}^0+{14}^{\prime }+{0,4}^{\prime }\\ & ={53}^0+{14}^{\prime }+0,4\times 1^{\prime }\\ & ={53}^0+{14}^{\prime }+0,4\times {60}^{″}\\ & ={53}^0+{14}^{\prime }+{24}^{″}\\ 53,{24}^0& ={53}^0{14}^{\prime }{24}^{″}\end{array}\\ & \begin{array}{rl}\text{b}.{23}^0{12}^{\prime }{24}^{″}& ={23}^0+12\times 1^{\prime }+24\times 1^{″}\\ & ={23}^0+12\times {\left(\frac{1}{60}\right)}^0+24\times {\left(\frac{1}{3600}\right)}^0\\ & ={23}^0+0,2^0+0,00{\overline{666}}^0\\ & =23,20{\overline{666}}^0\end{array}\end{array}$.

$\begin{array}{l}\\ 3.& \text{Ubahlah ke dalam sudut-sudut berikut dalam radian}\\ & \begin{array}{l}\\ \text{a}.& 3^0& \text{e}.& {90}^0& \text{i}.& {210}^0& \text{m}.& {300}^0\\ \text{b}.& {15}^0& \text{f}.& {120}^0& \text{j}.& {225}^0& \text{n}.& {315}^0\\ \text{c}.& {30}^0& \text{g}.& {135}^0& \text{k}.& {240}^0& \text{0}.& {12}^0{24}^{\prime }\\ \text{d}.& {60}^0& \text{h}.& {150}^0& \text{l}.& {270}^0& \text{p}.& {30}^0{12}^{\prime }{30}^{″}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Ingat bahwa}:{180}^0=180\times 1^0=\pi rad\Rightarrow 1^0={\displaystyle \frac{\pi }{180}rad}\\ & \begin{array}{|ll|}\hline \begin{array}{rl}\text{a}.3^0& =\cdots \pi rad\\ 3^0& ={\displaystyle \frac{3}{180}\pi rad}\\ & ={\displaystyle \frac{1}{60}\pi rad}\end{array}& \begin{array}{rl}\text{m}.{300}^0& =\cdots \pi rad\\ {300}^0& ={\displaystyle \frac{300}{180}\pi rad}\\ & ={\displaystyle \frac{5}{3}\pi rad}\end{array}\\ \begin{array}{rl}\text{c}.{30}^0& =\cdots \pi rad\\ {30}^0& ={\displaystyle \frac{30}{180}\pi rad}\\ & ={\displaystyle \frac{1}{6}\pi rad}\\ & \\ & \\ & \\ & \end{array}& \begin{array}{rl}\text{o}.{12}^0{24}^{\prime }& =\cdots \pi rad\\ {12}^0{24}^{\prime }& ={\displaystyle \frac{12+\left({\displaystyle \frac{24}{60}}\right)}{180}\pi rad}\\ & ={\displaystyle \frac{12+0,4}{180}\pi rad}\\ & ={\displaystyle \frac{12,4}{180}\pi rad}\\ & ={\displaystyle \frac{31}{450}\pi rad}\end{array}\\ \begin{array}{rl}\text{f}.{120}^0& =\cdots \pi rad\\ {120}^0& ={\displaystyle \frac{120}{180}\pi rad}\\ & ={\displaystyle \frac{2}{3}\pi rad}\end{array}& \\ \hline\end{array}\end{array}$.

$\begin{array}{l}\\ 4.& \text{Ubahlah ke dalam sudut-sudut berikut dalam derajat}\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle \frac{1}{2}\pi rad}& \text{e}.& {\displaystyle \frac{4}{3}\pi rad}\\ \\ \text{b}.& {\displaystyle \frac{3}{5}\pi rad}& \text{f}.& 2rad\\ \\ \text{c}.& {\displaystyle \frac{2}{9}\pi rad}& \text{g}.& 12rad\\ \\ \text{d}.& {\displaystyle \frac{7}{12}\pi rad}& \text{h}.& 15\pi rad\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Ingat bahwa}:\pi rad={180}^0\Rightarrow 1rad={\displaystyle \frac{{180}^0}{\pi }}\\ & \begin{array}{|ll|}\hline \begin{array}{rl}\text{a}.{\displaystyle \frac{1}{2}\pi rad}& ={\displaystyle \frac{1}{2}\pi \times {\displaystyle \frac{{180}^0}{\pi }}}\\ & ={90}^0\end{array}& \begin{array}{rl}\text{b}.{\displaystyle \frac{3}{5}\pi rad}& ={\displaystyle \frac{3}{5}\pi \times {\displaystyle \frac{{180}^0}{\pi }}}\\ & ={108}^0\end{array}\\ \begin{array}{rl}\text{d}.{\displaystyle \frac{7}{12}\pi rad}& ={\displaystyle \frac{7}{12}\pi \times {\displaystyle \frac{{180}^0}{\pi }}}\\ & ={105}^0\\ & \\ & \\ & \\ & \end{array}& \begin{array}{rl}\text{f}.2rad& =2\times {\displaystyle \frac{{180}^0}{\pi }}\\ & =2\times {\displaystyle \frac{{180}^0}{\frac{22}{7}}}\\ & ={\displaystyle \frac{{1260}^0}{11}}\\ & =114,\overline{54}{}^0\end{array}\\ \hline\end{array}\end{array}$.

$\text{LATIHAN SOAL}$.

$\begin{array}{l}\\ 1.& \text{Ubahlah sudut berikut ke satuan }\\ & \text{yang diminta}\\ & \text{a}.{27}^0=.....\text{rad}\\ & \text{b}.28\text{rad}=...{.}^0\\ & \text{c}.31,{35}^{\circ }=....\text{(dalam sexagesimal)}\\ & \text{d}.{30}^{\circ }{24}^{\prime }{12}^{″}=....{}^0\end{array}$.

$\begin{array}{l}\\ 2.& \text{Ubahlah sudut berikut ke satuan }\\ & \text{yang diminta}\\ & \text{a}.{135}^0=.....\text{rad}\\ & \text{b}.6\text{rad}=...{.}^0\\ & \text{c}.23,{45}^{\circ }=....\text{(dalam sexagesimal)}\\ & \text{d}.{45}^{\circ }{50}^{\prime }{36}^{″}=....{}^0\end{array}$.

$\begin{array}{l}\\ 3.& \text{Silahkan dicoba sendiri soal yang belum}\\ & \text{dibahas}\end{array}$.