Lanjutan Persamaan Trigonometri

$\begin{aligned}&\textrm{f. Menentukan Nilai Perbandingan Trigonometri}\\ &\quad\textrm{pada Segitiga Siku-Siku} \end{aligned}$.

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

$\begin{array}{ll}\\ 1.&\textrm{Diketahui}\: \: \tan \theta =\displaystyle \frac{a}{x} \\ &\textrm{Tentukanlah nilai}\: \: \displaystyle \frac{x}{\sqrt{a^{2}+x^{2}}}\\\\ &\color{blue}\textrm{Jawab}:\\ &\textrm{Perhatikanlah gambar segitiga AOX berikut} \end{array}$

$.\: \qquad\begin{aligned}&\textrm{Dengan rumus Pythagoras dapatr ditentukan}\\ &\textrm{panjang ruas}\: \: \textrm{AX, yaitu}:\\ &AO^{2}+OX^{2} =AX^{2}\\ &\textrm{atau}\\ &AX^{2}=AO^{2}+OX^{2} \\ &AX=\sqrt{AO^{2}+OX^{2}}\\ &\qquad =\sqrt{x^{2}+a^{2}},\\ &\textrm{maka}\\ &\bullet \quad \sin \theta =\displaystyle \frac{a}{\sqrt{x^{2}+a^{2}}}\\ &\bullet \quad \cos \theta =\displaystyle \frac{x}{\sqrt{x^{2}+a^{2}}} \\ &\textrm{Jadi, nilai}\: \: \displaystyle \frac{x}{\sqrt{x^{2}+a^{2}}}=\color{red}\cos \theta \end{aligned}$.

$\begin{array}{ll}\\ 2.&\textrm{Jika}\: \: \sin \beta +\cos \beta =\displaystyle \frac{6}{5},\: \textrm{tentukanlah}\\ &\textrm{a}.\quad \sin \beta \cos \beta \\ &\textrm{b}.\quad \sin ^{3}\beta +\cos ^{3}\beta \\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}\textrm{a}.\quad&\sin \beta +\cos \beta=\displaystyle \frac{6}{5}\\ &\color{red}\textrm{saat masing-masing ruas dikuadratkan,}\\ &\textrm{maka}\\ &\left (\sin \beta +\cos \beta \right )^{2}=\left (\displaystyle \frac{6}{5} \right )^{2}\\ &\sin ^{2}\beta +2\sin \beta \cos \beta +\cos ^{2}\beta =\displaystyle \frac{36}{25}\\ &\sin ^{2}\beta +\cos ^{2}\beta +2\sin \beta \cos \beta=\displaystyle \frac{36}{25}\\ &1+2\sin \beta \cos \beta=\displaystyle \frac{36}{25}\\ &2\sin \beta \cos \beta=\displaystyle \frac{36}{25}-1\\ &2\sin \beta \cos \beta=\displaystyle \frac{36-25}{25}=\frac{11}{25}\\ &\sin \beta \cos \beta=\color{blue}\displaystyle \frac{11}{50} \end{aligned}\\ &\begin{aligned}\textrm{b}.\quad&\sin ^{3}\beta +\cos ^{3}\beta \\ &=\left ( \sin \beta +\cos \beta \right )\left ( \sin ^{2}\beta +\cos ^{2}\beta -\sin \beta \cos \beta \right )\\ &=\left ( \sin \beta +\cos \beta \right )\left ( 1 -\sin \beta \cos \beta \right )\\ &=\left ( \displaystyle \frac{6}{5} \right ).\left ( 1-\displaystyle \frac{11}{50} \right )\\ &=\left ( \displaystyle \frac{6}{5} \right ).\left ( \displaystyle \frac{50-11}{50} \right )\\ &=\left ( \displaystyle \frac{6}{5} \right ).\left ( \displaystyle \frac{39}{50} \right )\\ &=\displaystyle \color{blue} \frac{3\times 39}{5\times 25}=\frac{117}{125} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 3.&\textrm{Jika}\: \: \tan \alpha =\displaystyle \frac{1}{\sqrt{7}},\: \textrm{tentukanlah}\\ &\left ( \displaystyle \frac{\csc ^{2}\alpha -\sec ^{2}\alpha }{\csc ^{2}\alpha +\sec ^{2}\alpha} \right ) \\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}\textrm{Diket}&\textrm{ahui bahwa}:\\ \tan \alpha &=\displaystyle \frac{1}{\sqrt{7}},\: \: \color{red}\textrm{dan ingat juga bahwa}\\ \sec ^{2}\alpha &=\tan ^{2}\alpha +1=\left ( \displaystyle \frac{1}{\sqrt{7}} \right )^{2}+1=\frac{1}{7}+1=\frac{8}{7}\\ \color{red}\textrm{Demik}&\color{red}\textrm{ian juga},\: \color{black}\cot \alpha =\displaystyle \frac{1}{\tan \alpha } =\displaystyle \frac{1}{\left ( \frac{1}{\sqrt{7}} \right )}=\sqrt{7},\\ \textrm{maka},&\: \: \csc ^{2}\alpha =\cot ^{2}\alpha +1=\left ( \sqrt{7} \right )^{2}+1=7+1=8\\ \textrm{Selanj}&\textrm{utnya}\\ &\left ( \displaystyle \frac{\csc ^{2}\alpha -\sec ^{2}\alpha }{\csc ^{2}\alpha +\sec ^{2}\alpha} \right )=\left ( \displaystyle \frac{8-\displaystyle \frac{8}{7}}{8+\displaystyle \frac{8}{7}} \right )\\ &=\displaystyle \frac{\displaystyle \frac{56-8}{7}}{\displaystyle \frac{56+8}{7}} \\ &=\displaystyle \frac{48}{64}\\ &=\color{blue}\displaystyle \frac{3}{4} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 4.&\textrm{Jika}\: \: \beta\: \: \textrm{sudut lancip dan}\: \: \cos \beta =\displaystyle \frac{3}{5},\\ &\textrm{tentukan nilai dari}\: \: \displaystyle \frac{\sin \beta \tan \beta -1}{2\tan ^{2}\beta }\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}\textrm{Diketahui}&\\ \cos \beta &=\displaystyle \frac{3}{5}\Rightarrow \sin ^{2}\beta +\cos ^{2}\beta =1\\ \sin ^{2}\beta &+\cos ^{2}\beta =1\\ \sin \beta &=\sqrt{1-\cos ^{2}\beta}=\sqrt{1-\left ( \displaystyle \frac{3}{5} \right )^{2}}\\ &=\sqrt{1-\displaystyle \frac{9}{25}}=\sqrt{\displaystyle \frac{16}{25}}=\displaystyle \frac{4}{5}\\ \textrm{Sehingga}\: &\tan \beta =\displaystyle \frac{\sin \beta }{\cos \beta }=\frac{\displaystyle \frac{4}{5}}{\displaystyle \frac{3}{5}}=\frac{4}{3}\\ &\color{red}\displaystyle \frac{\sin \beta \tan \beta -1}{2\tan ^{2}\beta }\color{black}=\displaystyle \frac{\displaystyle \frac{4}{5}\times \frac{4}{3}-1}{2\left ( \displaystyle \frac{4}{3} \right )^{2}}\\ &\: \, \quad\quad\quad\quad\quad\quad =\displaystyle \frac{\displaystyle \frac{16}{15}-1}{\displaystyle \frac{32}{9}}=\displaystyle \frac{\displaystyle \frac{1}{15}}{\displaystyle \frac{32}{9}}=\displaystyle \frac{9}{32\times 15}\\ &\: \, \quad\quad\quad\quad\quad\quad =\displaystyle \frac{3}{32\times 5}\\ &\: \, \quad\quad\quad\quad\quad\quad =\color{blue}\displaystyle \frac{3}{160} \end{aligned} \end{array}$

DAFTAR PUSTAKA

1. Sukino. 2016. Matematika untuk SMA/MA Kelas XI Kelompok Peminatan Matematika dan Ilmu-Ilmu Alam. Jakarta: ERLANGGA