Perbandingan Trigonometri Sudut dalam Segitiga Siku-Siku

B. Perbandingan Trigonometri Suatu Sudut

B. 1 Perbandingan Trigonometri pada Segitiha Siku-Siku

Perhatikanlah ilustrasi  dari segitiga ABC siku-siku di C berikut

$\begin{array}{c}\sin \alpha ={\displaystyle \frac{BC}{AB}\iff \csc \alpha ={\displaystyle \frac{AB}{BC}={\displaystyle \frac{1}{\sin \alpha }}}}\\ \\ \cos \alpha ={\displaystyle \frac{AC}{AB}\iff \sec \alpha ={\displaystyle \frac{AB}{AC}={\displaystyle \frac{1}{\cos \alpha }}}}\\ \\ \tan \alpha ={\displaystyle \frac{BC}{AC}\iff \cot \alpha ={\displaystyle \frac{AC}{BC}={\displaystyle \frac{1}{\tan \alpha }}}}\end{array}$.

B. 2 Perbandingan Trigonometri untuk Sudut Istimewa.

$\begin{array}{|ccccccccc|}\hline {\alpha }^0& 0^0& {30}^0& {45}^0& {60}^0& {90}^0& {180}^0& {270}^0& {360}^0\\ \sin {\alpha }^0& 0& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}\sqrt{2}}& {\displaystyle \frac{1}{2}\sqrt{3}}& 1& 0& -1& 0\\ \cos {\alpha }^0& 1& {\displaystyle \frac{1}{2}\sqrt{3}}& {\displaystyle \frac{1}{2}\sqrt{2}}& {\displaystyle \frac{1}{2}}& 0& -1& 0& 1\\ \tan {\alpha }^0& 0& {\displaystyle \frac{1}{3}\sqrt{3}}& 1& \sqrt{3}& TD& 0& TD& 0\\ \hline\end{array}$.

B. 3. Tripel Pythagoras pada Segitiga Siku-Siku

$\begin{array}{|cccc|}\hline a& b& c=\sqrt{a^2+b^2}& \text{Tripel}\\ 3& 4& 5& (3,4,5)\\ 5& 12& 13& (5,12,13)\\ 7& 24& 25& (7,24,25)\\ 8& 15& 17& (8,15,17)\\ 9& 40& 41& (9,40,41)\\ 11& 60& 61& (11,60,61)\\ 12& 35& 37& (12,35,37)\\ 15& 112& 113& (15,112,113)\\ 16& 63& 65& (16,63,65)\\ 17& 144& 145& (17,144,145)\\ 19& 180& 181& (19,180,181)\\ 20& 21& 29& (20,21,29)\\ 21& 220& 221& (21,220,221)\\ 29& 420& 421& (29,420,421)\\ 30& 224& 226& (30,224,226)\\ 31& 480& 481& (31,480,481)\\ 33& 544& 545& (33,544,545)\\ 35& 612& 613& (35,612,613)\\ 37& 684& 685& (37,684,685)\\ 39& 760& 761& (39,760,761)\\ 41& 840& 841& (41,840,841)\\ 43& 924& 925& (43,924,925)\\ 45& 1012& 1013& (45,1012,1013)\\ 47& 1104& 1105& (47,1104,1105)\\ 48& 55& 73& (48,55,73)\\ 49& 1200& 1201& (49,1200,1201)\\ 51& 1300& 1301& (51,1300,1301)\\ 60& 63& 87& (60,63,87)\\ \hline\end{array}$.

$\text{CONTOH SOAL}$.

$\begin{array}{l}\\ 1.& \text{Tentukanlah nilai perbandingan}\sin \alpha \\ & \cos \alpha ,\tan \alpha \text{untuk segitiga berikut}\end{array}$.

$.\begin{array}{rl}& \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}\text{a}.& \text{Untuk sisi miringnya adalah}\\ & \begin{array}{rl}\text{Sisi miring}& =\sqrt{3^2+4^2}\\ & =\sqrt{9+16}\\ & =\sqrt{25}\\ & =5\end{array}\\ & \text{Sehingga}\\ & \sin \alpha ={\displaystyle \frac{3}{5},\cos \alpha =\frac{4}{5}\text{dan}\tan \alpha =\frac{3}{4}}\\ & \text{Dengan langkah sama}\end{array}\\ & \begin{array}{rl}\text{b}.& \text{Untuk sisi miringnya adalah}\\ & \begin{array}{rl}\text{Sisi miring}& =\sqrt{5^2+{12}^2}\\ & =\sqrt{25+144}\\ & =\sqrt{169}\\ & =13\end{array}\\ & \text{Sehingga}\\ & \sin \alpha =\frac{12}{13}\cos \alpha =\frac{5}{13}\text{dan}\tan \alpha =\frac{12}{5}.\end{array}\end{array}$.

$\begin{array}{l}\\ 2.& \text{Tentukanlah nilai dari}\\ & \begin{array}{ll}\text{a}.& (\tan {30}^0+\sin {30}^0)\cos {30}^0\\ \text{b}.& {(\tan {60}^0)}^2+4{(\sin {60}^0)}^2\\ \text{c}.& \tan {60}^0-\sin {60}^0-\tan {30}^0\\ \text{d}.& {\displaystyle \frac{1+\sin {30}^0}{\sin {30}^0}+\frac{\cos {30}^0}{1+\sin {30}^0}}\\ \text{e}.& {\displaystyle \frac{2\tan {30}^0}{1+{\tan }^2{30}^0}}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\end{array}$.

$.\begin{array}{|ll|}\hline \begin{array}{ll}\text{a}.& (\tan {30}^0+\sin {30}^0)\cos {30}^0\\ & ={\displaystyle (\frac{1}{3}\sqrt{3}+\frac{1}{2}).\frac{1}{2}\sqrt{3}}\\ & ={\displaystyle \frac{1}{2}+\frac{1}{4}\sqrt{3}}\end{array}& \begin{array}{ll}\text{b}.& {(\tan {60}^0)}^2+4{(\sin {60}^0)}^2\\ & {\displaystyle ={\left(\sqrt{3}\right)}^2+4{\left(\frac{1}{2}\sqrt{3}\right)}^2}\\ & {\displaystyle =3+3}\\ & =6\end{array}\\ \begin{array}{ll}\text{c}.& \tan {60}^0-\sin {60}^0-\tan {30}^0\\ & ={\displaystyle \sqrt{3}-\frac{1}{2}\sqrt{3}-\frac{1}{3}\sqrt{3}}\\ & ={\displaystyle \sqrt{3}-\frac{5}{6}\sqrt{3}}\\ & ={\displaystyle \frac{1}{6}\sqrt{3}}\end{array}& \begin{array}{ll}\text{d}.& {\displaystyle \frac{1+\sin {30}^0}{\sin {30}^0}+\frac{\cos {30}^0}{1+\sin {30}^0}}\\ & ={\displaystyle \frac{1+\frac{1}{2}}{\frac{1}{2}}+\frac{\frac{1}{2}\sqrt{3}}{1+\frac{1}{2}}}\\ & ={\displaystyle 3+\frac{1}{3}\sqrt{3}}\end{array}.\\ \begin{array}{ll}\text{e}.& {\displaystyle \frac{2\tan {30}^0}{1+{\tan }^2{30}^0}}\\ & ={\displaystyle \frac{2\times \frac{1}{3}\sqrt{3}}{1+{\left(\frac{1}{3}\sqrt{3}\right)}^2}}\\ & ={\displaystyle \frac{\frac{2}{3}\sqrt{3}}{1+\frac{1}{3}}}\\ & ={\displaystyle \frac{\frac{2}{3}\sqrt{3}}{\frac{4}{3}}}\\ & ={\displaystyle \frac{1}{2}\sqrt{3}}\end{array}.& \\ \hline\end{array}$.

$\begin{array}{l}\\ 3.& \text{Perhatikanlah ilustrasi berikut}\end{array}$

$.\begin{array}{rl}& \text{Jika Jarak antara kucing seorang pencatat}\\ & \text{dan kucing adalah 100 m, maka jarak}\\ & \text{Pencatat tersebut dengan seorang tentara}\\ & \text{sebagaimana gambar tersebut di atas adalah}?\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Perhatikan gambar di atas dengan diberikan}\\ & \text{tambahan keterangan sebagai berikut}\end{array}$.

$.\begin{array}{rl}& \text{Ditanya berpakah panjang jarak }(x+100)?\\ & \begin{array}{rl}& \text{Pilih}:y=y\\ & \iff x.\tan {60}^0=(x+100).\tan {30}^0\\ & \iff x.\sqrt{3}=(x+100)\frac{1}{3}\sqrt{3}\\ & \iff 3x=x+100\\ & \iff 2x=100\\ & \iff x=50\end{array}\\ & \text{Jadi}x+100=50+100=150\text{meter}.\end{array}$.

$\begin{array}{l}\\ 4.& \text{Tentukanlah perbandingan trigonometri}\\ & \mathrm{\angle }XOA\text{jika}A(3,5)\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Perhatikan ilustrasi berikut}\end{array}$.

$\begin{array}{rl}& \text{Dengan memandang ilustrasi gambar di atas}\\ & \text{kita mendapatkan}\mathrm{△}OA{A}^{\prime },\text{dengan menggunakan}\\ & \text{teorema pythagoras kita mendapatkan}\\ & \begin{array}{rl}O{A}^2& ={\left(O{A}^{\prime }\right)}^2+{\left(A{A}^{\prime }\right)}^2\\ & =3^2+5^2\\ & =9+25\\ & =34\\ OA& =\sqrt{34}\end{array}.\\ & \text{Sehingga akan didapatkan}\\ & \begin{array}{ll}\text{a}.& {\displaystyle \sin A^{\prime }OA=\frac{5}{\sqrt{34}}=\frac{5}{34}\sqrt{34}}\\ \text{b}.& {\displaystyle \cos A^{\prime }OA=\frac{3}{\sqrt{34}}=\frac{3}{34}\sqrt{34}}\\ \text{c}.& {\displaystyle \tan A^{\prime }OA=\frac{5}{3}}\end{array}.\end{array}$.

$\text{LATIHAN SOAL}$.

$\begin{array}{l}\\ 1.& \text{Tentukanlah nilai dari}\\ & \begin{array}{ll}\text{a}.& (\tan {60}^0+\sin {45}^0)\cos 0^0\\ \text{b}.& {(\tan {30}^0)}^2+4{(\sin {30}^0)}^2\\ \text{c}.& \tan {60}^0+\sin {60}^0+\tan {30}^0\\ \text{d}.& {\displaystyle \frac{1+\sin {45}^0}{\sin {30}^0}+\frac{\cos {30}^0}{1+\sin {45}^0}}\\ \text{e}.& {\displaystyle \frac{2\tan {60}^0}{1+{\tan }^2{60}^0}}\end{array}\end{array}$.

$\begin{array}{l}\\ 2.& \text{Tentukan nilai perbandingan trigonometrinya}\\ & \text{dari gambar segitiga berikut}\end{array}$.

$\begin{array}{l}\\ 3.& \text{Diketahui}\tan \alpha =0,75.\text{Tentukan nilai}\\ & \text{perbandingan trigonometri berikut}\\ & \begin{array}{ll}\text{a}.& \sin \alpha \\ \text{b}.& \cos \alpha \\ \text{c}.& \csc \alpha \\ \text{d}.& \sec \alpha \\ \text{e}.& \cot \alpha \\ \text{f}.& {\sin }^2\alpha +{\cos }^2\alpha \end{array}\end{array}$.

$\begin{array}{l}\\ 4.& \text{Jika}1-\cos \beta ={\displaystyle \frac{9}{25(1+\cos \beta )}\text{Tentukan}}\\ & \text{nilai dari}\tan \beta \end{array}$.

DAFTAR PUSTAKA

1. Kurnianingsih, S., Kuntarti, Sulistiyono. 2007. Matematika SMA dan MA untuk Kelas X Semester 2 Standar Isi 2006. Jakarta: ESIS.
2. Marwanta, dkk. 2013. Matematika SMA Kelas X. Jakarta: YUDHISTIRA.
3. Yuana. R.A., Indriyastuti. 2017. Persepktif Matematika untuk Kelas X SMA dan MA Kelompok Matematika Wajib. Solo: TIGA SERANGKAI PUSTAKA MANDIRI.