Persamaan Trigonometri Dasar

A. Persamaan Trigonometri Dasar

a.	Jika $\tan f(x)=\tan g(x)$ maka:
	1)	$f(x)=g(x)+k.360^\circ$ atau$f(x)=g(x)+k.2\pi$
	2)	$f(x)=(180^\circ -g(x))+k.360^\circ$ atau$f(x)=(\pi -g(x))+k.2\pi$
b.	Jika$\cos x=\cos \alpha$ maka:
	1)	$f(x)=g(x)+k.360^\circ$ atau$f(x)=g(x)+k.2\pi$
	2)	$f(x)=-g(x)+k.360^\circ$ atau$f(x)=-g(x)+k.2\pi$
c.	Jika$\sin x=\sin \alpha$ maka:
	$f(x)=g(x)+k.180^\circ$ atau$f(x)=g(x)+k.\pi$
k = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Contoh 1.
Tentukan himpunan penyelesaian dari persamaan $\sin x=\sin 20^\circ$ untuk $0^\circ \le x\le 360^\circ$.
Penyelesaian:
$\sin x=\sin 20^\circ$, diperoleh $f(x)=x$ dan $g(x)=20^\circ$ maka:
1) $f(x)=g(x)+k.360^\circ$
$x=20^\circ +k.360^\circ$
$k=0\to x=20^\circ +0\times 360^\circ =20^\circ$
2) $f(x)=(180^\circ -g(x))+k.360^\circ$
$x=(180^\circ -20^\circ )+k.360^\circ$
$x=160^\circ +k.360^\circ$
$k=0\to x=160^\circ +0\times 360^\circ =160^\circ$
HP =$\{20^\circ ,160^\circ \}$ atau HP =$\left\{ \frac{20^\circ }{180^\circ }\pi ,\frac{160^\circ }{180^\circ }\pi \right\}$ =$\left\{ \frac{1}{9}\pi ,\frac{8}{9}\pi \right\}$

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Contoh 2.
Tentukan himpunan penyelesaian dari persamaan $\cos x=\cos 37^\circ$ untuk $0^\circ \le x\le 360^\circ$
Penyelesaian:
$\cos x=\cos 37^\circ$ diperoleh $f(x)=x$ dan $g(x)=37^\circ$ maka:
1) $f(x)=g(x)+k.360^\circ$
$x=37^\circ +k.360^\circ$
$k=0\to x=37^\circ +0\times 360^\circ =37^\circ$
2) $f(x)=-g(x)+k.360^\circ$
$x=-37^\circ +k.360^\circ$
$k=1\to x=-37^\circ +1\times 360^\circ =323^\circ$
HP =$\{37^\circ ,323^\circ \}$

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Contoh 3.
Tentukan himpunan penyelesaian dari persamaan $\tan 2x=\tan 30^\circ$ untuk $0 < x < \frac{3}{2}\pi$.
Penyelesaian:
$0 < x < \frac{3}{2}\pi$
$0^\circ < x < \frac{3}{2}\times 180^\circ$
$0^\circ < x < 270^\circ$
$\tan 2x=\tan 30^\circ$ diperoleh $f(x)=2x$ dan $g(x)=30^\circ$ maka:
$\begin{align}\color{blue}f(x) &\color{blue}= g(x)+k.180^\circ \\ 2x &= 30^\circ +k.180^\circ \\ x &= 15^\circ +k.90^\circ \end{align}$
$k=0\to x=15^\circ +0\times 90^\circ =15^\circ$
$k=1\to x=15^\circ +1\times 90^\circ =105^\circ$
$k=2\to x=15^\circ +2\times 90^\circ =195^\circ$
HP =$\{15^\circ ,105^\circ ,195^\circ \}$
atau
HP = $\left\{ \frac{15^\circ }{180^\circ }\pi ,\frac{105^\circ }{180^\circ }\pi ,\frac{195^\circ }{180^\circ }\pi \right\}$ = $\left\{ \frac{1}{12}\pi, \frac{7}{12}\pi, \frac{13}{12}\pi \right\}$

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Contoh 4.
Tentukan himpunan penyelesaian dari persamaan $\sin 3x=\sin 39^\circ$ untuk $-180^\circ < x < 180^\circ$.
Penyelesaian:
$\sin 3x=\sin 39^\circ$ diperoleh $f(x)=3x$ dan $g(x)=39^\circ$ maka:
1) $f(x)=g(x)+k.360^\circ$
$\begin{align}3x &= 39^\circ +k.360^\circ \\ x &= 13^\circ +k.120^\circ \end{align}$
$k=-1\to x=13^\circ +(-1).120^\circ =-107^\circ$
$k=0\to x=13^\circ +0\times 120^\circ =13^\circ$
$k=1\to x=13^\circ +1\times 120^\circ =133^\circ$
2) $f(x)=(180^\circ -g(x))+k.360^\circ$
$\begin{align}3x &= (180^\circ -39^\circ )+k.360^\circ \\ 3x &= 141^\circ +k.360^\circ \\ x &= 47^\circ +k.120^\circ \end{align}$
$k=-1\to x=47^\circ +(-1).120^\circ =-73^\circ$
$k=0\to x=47^\circ +0\times 120^\circ =47^\circ$
$k=1\to x=47^\circ +1\times 120^\circ =167^\circ$
HP =$\{-107^\circ ,-73^\circ ,13^\circ ,47^\circ ,133^\circ ,167^\circ \}$

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Contoh 5.
Tentukan himpunan penyelesaian dari persamaan $2\cos 4x+1=0$ untuk $0 < x < \pi$.
Penyelesaian:
$\begin{align}2\cos 4x+1 &= 0 \\ 2\cos 4x &= -1 \\ \cos 4x &= -\frac{1}{2} \\ \cos 4x &= \cos 120^\circ \end{align}$
diperoleh $f(x)=4x$ dan $g(x)=120^\circ$
1) $f(x)=g(x)+k.360^\circ$
$\begin{align}4x &= 120^\circ +k.360^\circ \\ x &= 30^\circ +k.90^\circ \end{align}$
$k=0\to x=30^\circ +0\times 90^\circ =30^\circ$
$k=1\to x=30^\circ +1\times 90^\circ =120^\circ$
2) $f(x)=-g(x)+k.360^\circ$
$\begin{align}4x &= -120^\circ +k.360^\circ \\ x &= -30^\circ +k.90^\circ \end{align}$
$k=1\to x=-30^\circ +1\times 90^\circ =60^\circ$
$k=2\to x=-30^\circ +2\times 90^\circ =150^\circ$
HP =$\{30^\circ ,60^\circ ,120^\circ ,150^\circ \}$
atau
HP = $\left\{ \frac{30^\circ }{180^\circ }\pi ,\frac{60^\circ }{180^\circ }\pi ,\frac{120^\circ }{180^\circ }\pi ,\frac{150^\circ }{180^\circ }\pi \right\}$ = $\left\{ \frac{1}{6}\pi ,\frac{1}{3}\pi ,\frac{2}{3}\pi ,\frac{5}{6}\pi \right\}$

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Contoh 6.
Tentukan himpunan penyelesaian dari persamaan $6\tan (x+10^\circ )-2\sqrt{3}=0$ untuk $0^\circ \le x\le 360^\circ$.
Penyelesaian:
$\begin{align}6\tan (x+10^\circ )-2\sqrt{3} &= 0 \\ 6\tan (x+10^\circ ) &= 2\sqrt{3} \\ \tan (x+10^\circ ) &= \frac{2\sqrt{3}}{6} \\ \tan (x+10^\circ ) &= \frac{1}{3}\sqrt{3} \\ \tan (x+10^\circ ) &= 30^\circ \end{align}$
diperoleh $f(x)=x+10^\circ$ dan $g(x)=30^\circ$ maka:
$\begin{align}\color{blue}f(x) &\color{blue}= g(x)+k.180^\circ \\ x+10^\circ &= 30^\circ +k.180^\circ \\ x &= 30^\circ -10^\circ +k.180^\circ \\ x &= 20^\circ +k.180^\circ \end{align}$
$k=0\to x=20^\circ +0\times 180^\circ =20^\circ$
$k=1\to x=20^\circ +1\times 180^\circ =200^\circ$
HP =$\{20^\circ ,200^\circ \}$ atau HP = $\left\{ \frac{20^\circ }{180^\circ }\pi ,\frac{200^\circ }{180^\circ }\pi \right\}$ = $\left\{ \frac{1}{9}\pi ,\frac{10}{9}\pi \right\}$

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Contoh 7.
Tentukan himpunan penyelesaian dari persamaan $\sin (2x-50^\circ )=\frac{1}{2}$ untuk $0^\circ < x < 270^\circ$.
Penyelesaian:
$\begin{align}\sin (2x-50^\circ ) &= \frac{1}{2} \\ \sin (2x-50^\circ ) &= \sin 30^\circ \end{align}$
diperoleh $f(x)=2x-50^\circ$ dan $g(x)=30^\circ$ maka:
1) $f(x)=g(x)+k.360^\circ$
$\begin{align}2x-50^\circ &= 30^\circ +k.360^\circ \\ 2x &= 30^\circ +50^\circ +k.360^\circ \\ 2x &= 80^\circ +k.360^\circ \\ x &= 40^\circ +k.180^\circ \end{align}$
$k=0\to x=40^\circ +0\times 180^\circ =40^\circ$
$k=1\to x=40^\circ +1\times 180^\circ =220^\circ$
2) $f(x)=(180^\circ -g(x))+k.360^\circ$
$\begin{align}2x-50^\circ &= (180^\circ -30^\circ )+k.360^\circ \\ 2x-50^\circ &= 150^\circ +k.360^\circ \\ 2x &= 150^\circ +50^\circ +k.360^\circ \\ 2x &= 200^\circ +k.360^\circ \\ x &= 100^\circ +k.180^\circ \end{align}$
$k=0\to x=100^\circ +0\times 180^\circ =100^\circ$
HP =$\{40^\circ ,100^\circ ,220^\circ \}$ atau
HP = $\left\{ \frac{40^\circ }{180^\circ }\pi ,\frac{100^\circ }{180^\circ }\pi ,\frac{220^\circ }{180^\circ }\pi \right\}$ = $\left\{ \frac{2}{9}\pi ,\frac{5}{9}\pi ,\frac{11}{9}\pi \right\}$

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Contoh 8.
Tentukan himpunan penyelesaian dari persamaan $2\cos (2x+30^\circ )-\sqrt{3}=0$ untuk $0 < x \le 2\pi$.
Penyelesaian:
$\begin{align}2\cos (2x+30^\circ )-\sqrt{3} &= 0 \\ 2\cos (2x+30^\circ ) &= \sqrt{3} \\ \cos (2x+30^\circ ) &= \frac{\sqrt{3}}{2} \\ \cos (2x+30^\circ ) &= \cos 30^\circ \end{align}$
diperoleh $f(x)=2x+30^\circ$ dan $g(x)=30^\circ$ maka:
1) $f(x)=g(x)+k.360^\circ$
$\begin{align}2x+30^\circ &= 30^\circ +k.360^\circ \\ 2x &= 30^\circ -30^\circ +k.360^\circ \\ 2x &= k.360^\circ \\ x &= k.180^\circ \end{align}$
$k=1\to x=1\times 180^\circ =180^\circ$
$k=2\to x=2\times 180^\circ =360^\circ$
2) $f(x)=-g(x)+k.360^\circ$
$\begin{align}2x+30^\circ &= -30^\circ +k.360^\circ \\ 2x &= -30^\circ -30^\circ +k.360^\circ \\ 2x &= -60^\circ +k.360^\circ \\ x &= -30^\circ +k.180^\circ \end{align}$
$k=1\to x=-30^\circ +1\times 180^\circ =60^\circ$
$k=2\to x=-30^\circ +2\times 180^\circ =330^\circ$
HP =$\{60^\circ ,180^\circ ,330^\circ ,360^\circ \}$
atau
HP = $\left\{ \frac{60^\circ }{180^\circ }\pi ,\frac{180^\circ }{180^\circ }\pi ,\frac{330^\circ }{180^\circ }\pi ,\frac{360^\circ }{180^\circ }\pi \right\}$ = $\left\{ \frac{1}{3}\pi ,\pi ,\frac{11}{6}\pi ,2\pi \right\}$

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Contoh 9.
Tentukan himpunan penyelesaian dari persamaan $\cos 2x=\cos 3x$ untuk $0^\circ \le x\le 360^\circ$.
Penyelesaian:
$\cos 2x=\cos 3x$ diperoleh $f(x)=2x$ dan $g(x)=3x$ maka:
1) $f(x)=g(x)+k.360^\circ$
$\begin{align}2x &= 3x+k.360^\circ \\ 2x-3x &= k.360^\circ \\ -x &= k.360^\circ \\ x &= -k.360^\circ \end{align}$
$k=-1\to x=-(-1)\times 360^\circ =360^\circ$
$k=0\to x=0\times 360^\circ =0^\circ$
2) $f(x)=-g(x)+k.360^\circ$
$\begin{align}2x &= -3x+k.360^\circ \\ 2x+3x &= k.360^\circ \\ 5x &= k.360^\circ \\ x &= k.72^\circ \end{align}$
$k=0\to x=0\times 72^\circ =0^\circ$
$k=1\to x=1\times 72^\circ =72^\circ$
$k=2\to x=2\times 72^\circ =144^\circ$
$k=3\to x=3\times 72^\circ =216^\circ$
$k=4\to x=4\times 72^\circ =288^\circ$
$k=5\to x=5\times 72^\circ =360^\circ$
HP =$\{0^\circ ,72^\circ ,144^\circ ,216^\circ ,288^\circ ,360^\circ \}$

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Contoh 10.
Tentukan himpunan penyelesaian dari persamaan $\sin (x+30^\circ )=\cos 2x$ untuk $0^\circ \le x\le 360^\circ$.
Penyelesaian:
$\begin{align}\sin (x+30^\circ ) &= \cos 2x \\ \sin (x+30^\circ ) &= \sin (90^\circ -2x) \end{align}$
diperoleh$f(x)=x+30^\circ$ dan $g(x)=90^\circ -2x$ maka:
1) $f(x)=g(x)+k.360^\circ$
$\begin{align}x+30^\circ &= 90^\circ -2x+k.360^\circ \\ x+2x &= 90^\circ -30^\circ +k.360^\circ \\ 3x &= 60^\circ +k.360^\circ \\ x &= 20^\circ +k.120^\circ \end{align}$
$k=0\to x=20^\circ +0\times 120^\circ =20^\circ$
$k=1\to x=20^\circ +1\times 120^\circ =140^\circ$
$k=2\to x=20^\circ +2\times 120^\circ =260^\circ$
2) $f(x)=(180^\circ -g(x))+k.360^\circ$
$\begin{align}x+30^\circ &= (180^\circ -(90^\circ -2x))+k.360^\circ \\ x+30^\circ &= 90^\circ +2x+k.360^\circ \\ x-2x &= 90^\circ -30^\circ +k.360^\circ \\ -x &= 60^\circ +k.360^\circ \\ x &=-60^\circ -k.360^\circ \end{align}$
$k=-1\to x=-60^\circ -(-1).360^\circ =300^\circ$
HP =$\{20^\circ ,140^\circ ,260^\circ ,300^\circ \}$
atau
HP =$\left\{ \frac{20^\circ }{180^\circ }\pi ,\frac{140^\circ }{180^\circ }\pi ,\frac{260^\circ }{180^\circ }\pi ,\frac{300^\circ }{180^\circ }\pi \right\}$
HP =$\left\{ \frac{1}{9}\pi ,\frac{7}{9}\pi ,\frac{13}{9}\pi ,\frac{5}{3}\pi \right\}$

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B. Soal Latihan

Tentukan himpunan penyelesaian dari persamaan trigonometri berikut:

1. $\sin 3x=0$ untuk $0^\circ < x < 360^\circ$.
2. $2\cos (2x-60^\circ )-\sqrt{3}=0$ untuk $0\le x\le 2\pi$.
3. $\tan (x-45^\circ )=\cot 60^\circ$ untuk $0\le x\le 2\pi$.
4. $\cos 2x-\cos x=0$ untuk $0\le x\le 2\pi$.
5. $\sin 3x=\cos 2x$ untuk $0\le x\le 2\pi$.

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