Koordinat Kartesius dan Koordinat Kutub serta Kuadran

C. Koordinat Kartesius dan Koordinat Kutub/Polar

Perhatikan ilustrasi berikut

$\begin{array}{|c|c|}\hline  \textrm{Kartesius}\: \rightarrow \: \textrm{Kutub}&\textrm{Kutub}\: \rightarrow \: \textrm{Kartesius}\\\hline P(x,y)\: \rightarrow \: P\left ( r,\alpha ^{0} \right )&P\left ( r,\alpha ^{0} \right )\: \rightarrow \: P(x,y)\\\hline \begin{aligned}&r=\sqrt{x^{2}+y^{2}},\\ &\displaystyle \tan \alpha ^{0}=\frac{y}{x},\quad \displaystyle \alpha ^{0}=\arctan \frac{y}{x} \end{aligned}&\left\{\begin{matrix} x=r.\cos \alpha ^{0}\\ \\ y=r.\sin \alpha ^{0} \end{matrix}\right.\\\hline \end{array}$.

D. Perbandingan Trigonometri di Berbagai Kuadran

D. 1 Untuk Sudut 0 sampai dengan 360 derajat.

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

$\begin{array}{llll} (1).&\textrm{Perbandingan Trigonometri untuk Sudut}\: \left ( 90^{0}-\alpha \right )\\ &\textrm{a}.\quad \sin \alpha \rightarrow \sin \left ( 90^{0}-\theta \right )=\cos \theta \\ &\textrm{b}.\quad \cos \alpha \rightarrow \cos \left ( 90^{0}-\theta \right )=\sin \theta \\ &\textrm{c}.\quad \tan \alpha \rightarrow \tan \left ( 90^{0}-\theta \right )=\cot \theta \\ &\textrm{d}.\quad \cot \alpha \rightarrow \cot \left ( 90^{0}-\theta \right )=\tan \theta \end{array}$.

$\begin{array}{llll} (2).&\textrm{Perbandingan Trigonometri untuk Sudut}\: \left ( 180^{0}-\alpha \right )\\ &\textrm{a}.\quad \sin \alpha \rightarrow \sin \left ( 180^{0}-\theta \right )=\sin \theta \\ &\textrm{b}.\quad \cos \alpha \rightarrow \cos \left ( 180^{0}-\theta \right )=-\cos \theta \\ &\textrm{c}.\quad \tan \alpha \rightarrow \tan \left ( 180^{0}-\theta \right )=-\tan \theta \\ &\textrm{d}.\quad \cot \alpha \rightarrow \cot \left ( 180^{0}-\theta \right )=-\cot \theta \end{array}$.

$\begin{array}{llll} (3).&\textrm{Perbandingan Trigonometri untuk Sudut}\: \left ( 180^{0}+\alpha \right )\\ &\textrm{a}.\quad \sin \alpha \rightarrow \sin \left ( 180^{0}+\theta \right )=-\sin \theta \\ &\textrm{b}.\quad \cos \alpha \rightarrow \cos \left ( 180^{0}+\theta \right )=-\cos \theta \\ &\textrm{c}.\quad \tan \alpha \rightarrow \tan \left ( 180^{0}+\theta \right )=\tan \theta \\ &\textrm{d}.\quad \cot \alpha \rightarrow \cot \left ( 180^{0}+\theta \right )=\cot \theta \end{array}$.

$\begin{array}{llll} (4).&\textrm{Perbandingan Trigonometri untuk Sudut}\: \left ( 360^{0}-\alpha \right )\\ &\textrm{a}.\quad \sin \alpha \rightarrow \sin \left ( 360^{0}-\theta \right )=-\sin \theta \\ &\textrm{b}.\quad \cos \alpha \rightarrow \cos \left ( 360^{0}-\theta \right )=\cos \theta \\ &\textrm{c}.\quad \tan \alpha \rightarrow \tan \left ( 360^{0}-\theta \right )=-\tan \theta \\ &\textrm{d}.\quad \cot \alpha \rightarrow \cot \left ( 360^{0}-\theta \right )=-\cot \theta \end{array}$.

D. 2 Untuk Sudut yang Lain

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

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

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

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

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

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