Lanjutan 2 Persamaan Trigonometri

 A. 2  Relasi Sudut

Mengingatkan kembali materi tentang nilai sudut diberbagai kuadran yang selanjutnya berkaitan erat dengan relasi sudutnya dari kuadran selain satu diubah ke kuadran satu supaya mudah menentukan nilai trigonometri.

Untuk tanda perbandingan trigonometrinya berkaitan dengan relasi sudutnya adalah disajikan sebagaimana dalam bagan berikut

$\begin{array}{ccc|cccc} \textrm{Nilai yang positif}&&&&&\\ \textrm{hanya}\quad \color{red}\textbf{sinus}&&&&\textrm{Semua nilai trigon}&\color{blue}\textbf{positif}\\ &&&&&\\\hline \textrm{Nilai yang positif}&&&&\textrm{Nilai yang positif}&\\ \textrm{hanya}\quad \color{red}\textbf{tangen}&&&&\textrm{hanya}\quad \color{red}\textbf{cosinus}&\\ \end{array}$.

atau

$\begin{array}{ccc|cccc} \begin{array}{ll}\\ \begin{cases} \color{blue}\sin & =+ \\ \cos & =- \\ \tan & =- \\ \color{red}\csc & = +\\ \sec & = -\\ \cot & = - \end{cases}& \end{array}&&&&&\begin{array}{ll}\\ \begin{cases} \color{blue}\sin & =+ \\ \color{blue}\cos & =+ \\ \color{blue}\tan & =+ \\ \color{red}\csc & = +\\ \color{red}\sec & = +\\ \color{red}\cot & = + \end{cases}& \end{array}\\ &&&&\\ &&&&&\\\hline &&&&&\\ \begin{array}{ll}\\ \begin{cases} \sin & =- \\ \cos & =- \\ \color{blue}\tan & =+ \\ \csc & = -\\ \sec & = -\\ \color{red}\cot & = + \end{cases}& \end{array}&&&&&\begin{array}{ll}\\ \begin{cases} \sin & =- \\ \color{blue}\cos & =+ \\ \tan & =- \\ \csc & = -\\ \color{red}\sec & = +\\ \cot & = - \end{cases}& \end{array}\\ \end{array}$.

Adapun penjabaran sudut-sudut yang berelasi sebagaimana ilustrasi bagan berikut, yaitu:

$\begin{array}{ccc|cccc} \textrm{Kuadran II}&&&&\textrm{Kuadran I}&\\ \left (180^{\circ}-\alpha \right )&&&&\textrm{Semua nilai trigon}&\color{blue}\textbf{positif}\\ &&&&&\\\hline \textrm{Kuadran III}&&&&\textrm{Kuadran IV}&\\ \left (180^{\circ}+\alpha \right )&&&&\left (360^{\circ}-\alpha \right )& \\ \end{array}$

Ketentuan perubahan trigonometri berkaitan dengan sudut berelasi adalah sebagaimana tabel berikut:

KUADRAN PERTAMA

$\begin{array}{|c|c|l|}\hline \textrm{Posisi}&\textrm{Perubahan}&\qquad\textrm{Relasi Sudut}\\\hline \begin{aligned}&\textrm{Kuadran I}\\ &0^{\circ}<\alpha <90^{\circ}\\ &=\left ( 90^{\circ}-\alpha \right ) \end{aligned}&\begin{cases} \color{blue}\sin & =\cos \\ \cos & =\sin \\ \tan & =\cot \\ \color{red}\csc & = \sec \\ \sec & = \csc \\ \cot & = \tan \end{cases}&\begin{aligned}\sin \left ( 90^{\circ}-\alpha \right )&=\cos \alpha \\ \cos \left ( 90^{\circ}-\alpha \right )&=\sin \alpha \\ \tan \left ( 90^{\circ}-\alpha \right )&=\cot \alpha\\ \csc \left ( 90^{\circ}-\alpha \right )&=\sec \alpha \\ \sec \left ( 90^{\circ}-\alpha \right )&=\csc \alpha\\ \cot \left ( 90^{\circ}-\alpha \right )&=\tan \alpha \end{aligned}\\\hline \end{array}$.

KUADRAN KEDUA

ada 2 pilihan yaitu:

pertama

$\begin{array}{|c|c|l|}\hline \textrm{Posisi}&\textrm{Perubahan}&\qquad\textrm{Relasi Sudut}\\\hline \begin{aligned}&\textrm{Kuadran II}\\ &90^{\circ}<\alpha <180^{\circ}\\ &=\left ( 90^{\circ}+\alpha \right ) \end{aligned}&\begin{cases} \color{blue}\sin & =\cos \\ \cos & =\sin \\ \tan & =\cot \\ \color{red}\csc & = \sec \\ \sec & = \csc \\ \cot & = \tan \end{cases}&\begin{aligned}\sin \left ( 90^{\circ}+\alpha \right )&=\cos \alpha \\ \cos \left ( 90^{\circ}+\alpha \right )&=-\sin \alpha \\ \tan \left ( 90^{\circ}+\alpha \right )&=-\cot \alpha\\ \csc \left ( 90^{\circ}+\alpha \right )&=\sec \alpha \\ \sec \left ( 90^{\circ}+\alpha \right )&=-\csc \alpha\\ \cot \left ( 90^{\circ}+\alpha \right )&=-\tan \alpha \end{aligned}\\\hline \end{array}$.

kedua

$\begin{array}{|c|c|l|}\hline \textrm{Posisi}&\textrm{Tidak Ada Perubahan}&\qquad\textrm{Relasi Sudut}\\\hline \begin{aligned}&\textrm{Kuadran II}\\ &90^{\circ}<\alpha <180^{\circ}\\ &=\left ( 180^{\circ}-\alpha \right ) \end{aligned}&\begin{cases} \color{blue}\sin & =\sin \\ \cos & =\cos \\ \tan & =\tan \\ \color{red}\csc & = \csc \\ \sec & = \sec \\ \cot & = \cot \end{cases}&\begin{aligned}\sin \left ( 180^{\circ}-\alpha \right )&=\sin \alpha \\ \cos \left ( 180^{\circ}-\alpha \right )&=-\cos \alpha \\ \tan \left ( 180^{\circ}-\alpha \right )&=-\tan \alpha\\ \csc \left ( 180^{\circ}-\alpha \right )&=\csc \alpha \\ \sec \left ( 180^{\circ}-\alpha \right )&=-\sec \alpha\\ \cot \left ( 180^{\circ}-\alpha \right )&=-\cot \alpha \end{aligned}\\\hline \end{array}$.

KUADRAN KETIGA

ada 2 pilihan juga yaitu:

pertama

$\begin{array}{|c|c|l|}\hline \textrm{Posisi}&\textrm{Tidak Ada Perubahan}&\qquad\textrm{Relasi Sudut}\\\hline \begin{aligned}&\textrm{Kuadran III}\\ &180^{\circ}<\alpha <270^{\circ}\\ &=\left ( 180^{\circ}+\alpha \right ) \end{aligned}&\begin{cases} \color{blue}\sin & =\sin \\ \cos & =\cos \\ \tan & =\tan \\ \color{red}\csc & = \csc \\ \sec & = \sec \\ \cot & = \cot \end{cases}&\begin{aligned}\sin \left ( 180^{\circ}+\alpha \right )&=-\sin \alpha \\ \cos \left ( 180^{\circ}+\alpha \right )&=-\cos \alpha \\ \tan \left ( 180^{\circ}+\alpha \right )&=\tan \alpha\\ \csc \left ( 180^{\circ}+\alpha \right )&=-\csc \alpha \\ \sec \left ( 180^{\circ}+\alpha \right )&=-\sec \alpha\\ \cot \left ( 180^{\circ}+\alpha \right )&=\cot \alpha \end{aligned}\\\hline \end{array}$.

kedua

$\begin{array}{|c|c|l|}\hline \textrm{Posisi}&\textrm{Perubahan}&\qquad\textrm{Relasi Sudut}\\\hline \begin{aligned}&\textrm{Kuadran III}\\ &180^{\circ}<\alpha <270^{\circ}\\ &=\left ( 270^{\circ}-\alpha \right ) \end{aligned}&\begin{cases} \color{blue}\sin & =\cos \\ \cos & =\sin \\ \tan & =\cot \\ \color{red}\csc & = \sec \\ \sec & = \csc \\ \cot & = \tan \end{cases}&\begin{aligned}\sin \left ( 270^{\circ}-\alpha \right )&=-\cos \alpha \\ \cos \left ( 270^{\circ}-\alpha \right )&=-\sin \alpha \\ \tan \left ( 270^{\circ}-\alpha \right )&=\cot \alpha\\ \csc \left ( 270^{\circ}-\alpha \right )&=-\sec \alpha \\ \sec \left ( 270^{\circ}-\alpha \right )&=-\csc \alpha\\ \cot \left ( 270^{\circ}-\alpha \right )&=\tan \alpha \end{aligned}\\\hline \end{array}$.

KUADRAN KEEMPAT

ada 2 pilihan juga yaitu:

pertama

$\begin{array}{|c|c|l|}\hline \textrm{Posisi}&\textrm{Perubahan}&\qquad\textrm{Relasi Sudut}\\\hline \begin{aligned}&\textrm{Kuadran IV}\\ &270^{\circ}<\alpha <360^{\circ}\\ &=\left ( 270^{\circ}+\alpha \right ) \end{aligned}&\begin{cases} \color{blue}\sin & =\cos \\ \cos & =\sin \\ \tan & =\cot \\ \color{red}\csc & = \sec \\ \sec & = \csc \\ \cot & = \tan \end{cases}&\begin{aligned}\sin \left ( 270^{\circ}+\alpha \right )&=-\cos \alpha \\ \cos \left ( 270^{\circ}+\alpha \right )&=\sin \alpha \\ \tan \left ( 270^{\circ}+\alpha \right )&=-\cot \alpha\\ \csc \left ( 270^{\circ}+\alpha \right )&=-\sec \alpha \\ \sec \left ( 270^{\circ}+\alpha \right )&=\csc \alpha\\ \cot \left ( 270^{\circ}+\alpha \right )&=-\tan \alpha \end{aligned}\\\hline \end{array}$.

kedua

$\begin{array}{|c|c|l|}\hline \textrm{Posisi}&\textrm{Tidak Ada Perubahan}&\qquad\textrm{Relasi Sudut}\\\hline \begin{aligned}&\textrm{Kuadran IV}\\ &270^{\circ}<\alpha <360^{\circ}\\ &=\left ( 360^{\circ}-\alpha \right ) \end{aligned}&\begin{cases} \color{blue}\sin & =\sin \\ \cos & =\cos \\ \tan & =\tan \\ \color{red}\csc & = \csc \\ \sec & = \sec \\ \cot & = \cot \end{cases}&\begin{aligned}\sin \left ( 360^{\circ}-\alpha \right )&=-\sin \alpha \\ \cos \left ( 360^{\circ}-\alpha \right )&=\cos \alpha \\ \tan \left ( 360^{\circ}-\alpha \right )&=-\tan \alpha\\ \csc \left ( 360^{\circ}-\alpha \right )&=-\csc \alpha \\ \sec \left ( 360^{\circ}-\alpha \right )&=\sec \alpha\\ \cot \left ( 360^{\circ}-\alpha \right )&=-\cot \alpha \end{aligned}\\\hline \end{array}$.

 A. 3  Sudut Negatif dan Sudut lebih Besar dari  $360^{\circ}$

$\begin{aligned}\textrm{a}.\quad&\begin{cases} \sin \left ( -A \right ) & =-\sin A \\ \cos \left ( -A \right ) & =\cos A \\ \tan \left ( -A \right ) & = -\tan A \end{cases}\\ \textrm{b}.\quad&\begin{cases} \csc \left ( -A \right ) &=-\csc A \\ \sec \left ( -A \right ) &=\sec A \\ \cot \left ( -A \right ) &=-\cot A \end{cases}\\ \textrm{c}.\quad&\begin{cases} \sin \left ( n.360^{\circ}+A \right ) & =\sin A \\ \cos \left ( n.360^{\circ}+A \right ) & =\cos A \\ \tan \left ( n.360^{\circ}+A \right ) & =\tan A \end{cases},\qquad n\in \mathbb{N} \end{aligned}$.

Catatan : $0^{\circ}$=$360^{\circ}$=$720^{\circ}$=$1080^{\circ}$=$n.360^{\circ}$

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

$\begin{array}{ll}\\ 1.&\textrm{Tentukanlah nilai}\\ &\textrm{a}.\quad\sin 120^{\circ}\\ &\textrm{b}.\quad\cos 240^{\circ}\\ &\textrm{c}.\quad\tan 315^{\circ}\\\\ &\textrm{Jawab}:\\ &\begin{aligned}\textrm{a}.\quad\sin 120^{\circ}&=\sin \left ( 180^{\circ}-60^{\circ} \right )=\sin 60^{\circ}\\ &=\displaystyle \frac{1}{2}\sqrt{3},\qquad \color{red}\textrm{atau}\\ &=\sin \left ( 90^{\circ}+30^{\circ} \right )=\cos 30^{\circ}=\displaystyle \frac{1}{2}\sqrt{3}\\ \textrm{b}.\quad\cos 240^{\circ}&=\cos \left ( 180^{\circ}+60^{\circ} \right) =-\cos 60^{\circ}\\ &=-\displaystyle \frac{1}{2},\qquad \color{red}\textrm{atau}\\ &=\cos \left ( 270^{\circ}-30^{\circ} \right )=-\sin 30^{\circ}=-\frac{1}{2}\\ \textrm{c}.\quad\tan 315^{\circ}&=\tan \left ( 360^{\circ}-45^{\circ} \right )=-\tan 45^{\circ}\\ &=-1,\qquad \color{red}\textrm{atau}\\ &=\tan \left ( 270^{\circ}+45^{\circ} \right )=-\cot 45^{\circ}=-1 \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 2.&\textrm{Buktikan bahwa}\\\\ &\textrm{a}.\quad \displaystyle \frac{\cos \left ( 90^{\circ}-B \right )}{\sec B}+\frac{\sin \left ( 90^{\circ}-B \right )}{\csc B}=2\sin B\cos B\\\\ &\textrm{b}.\quad \tan C+\tan \left ( 90^{\circ}-C \right )=\sec C.\sec \left ( 90^{\circ}-C \right )\\\\ &\textbf{Bukti}:\\ &\begin{aligned}\textrm{a}.\quad&\displaystyle \frac{\cos \left ( 90^{\circ}-B \right )}{\sec B}+\frac{\sin \left ( 90^{\circ}-B \right )}{\csc B}\\ &=\displaystyle \frac{\sin B}{\sec B}+\frac{\cos B}{\csc B}\\ &=\displaystyle \frac{\sin B}{\displaystyle \frac{1}{\cos B}}+\frac{\cos B}{\displaystyle \frac{1}{\sin B}}\\ &=\sin B\cos B+\sin B\cos B\\ &=2\sin B\cos B\qquad\quad \blacksquare \end{aligned} \\ &\begin{aligned}\textrm{b}.\quad&\tan C+\tan \left ( 90^{\circ}-C \right )\\ &=\tan C+\cot C\\ &=\displaystyle \frac{\sin C}{\cos C}+\frac{\cos C}{\sin C}\\ &=\displaystyle \frac{\sin ^{2}C+\cos ^{2}C}{\sin C\cos C}=\displaystyle \frac{1}{\sin C\cos C}\\ &=\displaystyle \frac{1}{\cos C}.\frac{1}{\sin C}\\ &=\sec C.\csc C\\ &=\sec C.\sec \left ( 90^{\circ}-C \right )\qquad\quad \blacksquare \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 3.&\textrm{Tentukanlah nilai}\\ &\textrm{a}.\quad \tan \left ( A-90^{\circ} \right )\sin \left ( -A \right )\\ &\textrm{b}.\quad\cos 540^{\circ}+\sin 690^{\circ}\\ &\textrm{c}.\quad \sin 2021^{\circ}+\cos 2021^{\circ}\\\\ &\textrm{Jawab}:\\ &\begin{aligned}\textrm{a}.\quad&\tan \left ( A-90^{\circ} \right )\sin \left ( -A \right )\\ &=\tan \left ( -\left (90^{\circ}-A \right ) \right )\left ( -\sin A \right )\\ &=-\tan \left ( 90^{\circ}-A \right )\left ( -\sin A \right )\\ &= \tan \left ( 90^{\circ}-A \right )\left ( \sin A \right )\\ &=\cot A.\sin A\\ &=\displaystyle \frac{\cos A}{\sin A}.\sin A\\ &=\cos A \end{aligned} \\ &\begin{aligned}\textrm{b}.\quad&\cos 540^{\circ}+\sin 690^{\circ}\\ &=\cos \left ( 360^{\circ}+180^{\circ} \right )+\sin \left ( 720^{\circ}-30^{\circ} \right )\\ &=\cos \left ( 0^{\circ}+180^{\circ} \right )+\sin \left ( 0^{\circ}-30^{\circ} \right )\\ &=\cos 180^{\circ}+ \sin \left ( -30^{\circ} \right ) \\ &=\cos 180^{\circ}-\sin 30^{\circ}\\ &=-1-\displaystyle \frac{1}{2}\\ &=-\displaystyle \frac{3}{2} \end{aligned} \\ &\begin{aligned}\textrm{c}.\quad&\sin 2021^{\circ}+\cos 2021^{\circ}\\ &=\sin \left ( 5.360^{\circ}+221^{\circ} \right )+\cos \left ( 5.360^{\circ}+221^{\circ} \right )\\ &=\sin \left (0^{\circ}+221^{\circ} \right )+\cos \left (0^{\circ}+221^{\circ} \right )\\ &=\sin 221^{\circ}+\cos 221^{\circ}\\ &=\sin \left ( 180^{\circ}+41^{\circ} \right )+\cos \left ( 180^{\circ}+41^{\circ} \right )\\ &=-\sin 41^{\circ}-\cos 41^{\circ} \end{aligned} \end{array}$