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  $\angle AOB$.

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.

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

Perhatikan ilustrasi berikut

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

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

$\LARGE\colorbox{yellow}{CONTOH SOAL}$.

$\begin{array}{ll}\\ 1.&\textrm{Ubahlah sudut berikut ke satuan}\\ & \textrm{yang diminta}\\ &\textrm{a}.\quad 1^{\circ}=....\: \textrm{rad}\\ &\textrm{b}.\quad 1\: \textrm{rad}=....\: ^{0}\\\\ &\textbf{Jawab}:\\ &\begin{aligned}& \begin{aligned}\textrm{a}.\quad\qquad 360^{0}&=2\pi \: \textrm{rad}\\ 360\times 1^{0}&=2\pi \: \textrm{rad}\\ 1^{0}&=\displaystyle \frac{2\pi }{360}\\ &=\displaystyle \frac{\pi }{180}\: \textrm{rad} \end{aligned}\\&\begin{aligned}\textrm{b}.\quad\qquad 2\pi \: \textrm{rad}&=360^{0}\\ 2\pi \times 1\: \textrm{rad}&=360^{0}\\ 1\: \textrm{rad}&=\left ( \displaystyle \frac{360}{2\pi } \right )^{0}\\ &=\left ( \displaystyle \frac{180}{\pi } \right )^{0} \end{aligned} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 2.&\textrm{Ubahlah sudut berikut ke satuan }\\ &\textrm{yang diminta}\\&\textrm{a}.\quad 53,24^{\circ}=....\: \textrm{(dalam sexagesimal)}\\ &\textrm{b}.\quad 23^{\circ}{12}'{24}''=....\: ^{0}\\\\&\textbf{Jawab}:\\&\begin{aligned}\textrm{a}.\quad53,24^{0}&=53^{0}+0,24^{0}\\ &=53^{0}+0,24\times 1^{0}\\ &=53^{0}+0,24\times {60}'\\ &=53^{0}+{14,4}'\\ &=53^{0}+{14}'+{0,4}'\\ &=53^{0}+{14}'+0,4\times {1}'\\ &=53^{0}+{14}'+0,4\times {60}''\\ &=53^{0}+{14}'+{24}''\\ 53,24^{0}&=53^{0}{14}'{24}'' \end{aligned}\\ &\begin{aligned}\textrm{b}.\quad 23^{0}{12}'{24}''&=23^{0}+12\times {1}'+24\times {1}''\\ &=23^{0}+12\times \left ( \frac{1}{60} \right )^{0}+24\times \left ( \frac{1}{3600} \right )^{0}\\&=23^{0}+0,2^{0}+0,00\overline{666}^{0}\\ &=23,20\overline{666}^{0}\end{aligned} \end{array}$.

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

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

$\LARGE\colorbox{yellow}{LATIHAN SOAL}$.

$\begin{array}{ll}\\ 1.&\textrm{Ubahlah sudut berikut ke satuan }\\ &\textrm{yang diminta}\\ &\textrm{a}.\quad27^{0}=.....\: \textrm{rad}\\ &\textrm{b}.\quad 28\: \textrm{rad}=....^{0}\\  &\textrm{c}.\quad 31,35^{\circ}=....\: \textrm{(dalam sexagesimal)}\\ &\textrm{d}.\quad 30^{\circ}{24}'{12}''=....\: ^{0} \end{array}$.

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

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