Lanjutan 3 Persamaan Trigonometri

 B. 1 Persamaan Trigonometri Sederhana

Dalam penyelesaian persamaan trigonometri sederhana dapat digunakan salah satu rumus berikut, yaitu:

$\begin{aligned}(1).\quad\sin x&=\sin \alpha ^{\circ}\\ &\left\{\begin{matrix} x=\alpha ^{\circ}+k.360^{\circ}\qquad\qquad\\ \color{red}\textrm{atau}\qquad\qquad\\ x=\left ( 180^{\circ}-\alpha ^{\circ} \right )+k.360^{\circ} \end{matrix}\right.\\ (2).\quad\cos x&=\cos \alpha ^{\circ}\\ &\left\{\begin{matrix} x=\alpha ^{\circ}+k.360^{\circ}\: \: \: \\ \color{red}\textrm{atau}\\ x=-\alpha ^{\circ}+k.360^{\circ} \end{matrix}\right.\\ (3).\quad\tan x&=\tan \alpha ^{\circ}\\ &x=\alpha ^{\circ}+k.180^{\circ} \end{aligned}$.

Jika sudutnya dinyatakan dalam phi radian $\left (\pi \quad \textrm{dibaca}:\: \: phi \right )$, maka persamaan trigonometri sederhananya adalah:

$\begin{aligned}(1).\quad\sin x&=\sin \alpha ^{\circ}\\ &\left\{\begin{matrix} x=\alpha ^{\circ}+k.2\pi \qquad\quad\\ \color{red}\textrm{atau}\qquad\qquad\\ x=\left ( \pi -\alpha ^{\circ} \right )+k.2\pi \end{matrix}\right.\\ (2).\quad\cos x&=\cos \alpha ^{\circ}\\ &\left\{\begin{matrix} x=\alpha ^{\circ}+k.2\pi \: \: \: \\ \color{red}\textrm{atau}\\ x=-\alpha ^{\circ}+k.2\pi \end{matrix}\right.\\ (3).\quad\tan x&=\tan \alpha ^{\circ}\\ &x=\alpha ^{\circ}+k.\pi \end{aligned}$.

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

$\begin{array}{ll}\\ 1.&\textrm{Tentukanlah akar-akar persamaan trigonometri}\\ &\textrm{berikut dan tentukan pula himpunan}\\ &\textrm{penyelesaiannya untuk}\: \: 0^{\circ}\leq x\leq 360^{\circ}\\ &\textrm{a}.\quad \sin x=\sin 50^{\circ}\\ &\textrm{b}.\quad \cos x=\cos 50^{\circ}\\ &\textrm{c}.\quad \tan x=\tan 50^{\circ}\\\\ &\textbf{Jawab}:\\ &\color{black}\begin{aligned}.\: \quad\textrm{a}.\quad\sin x&=\sin 50^{\circ}\\ x&=\begin{cases} 50^{\circ} & +k.360^{\circ}\\ \left (180^{\circ}-50^{\circ} \right ) & +k.360^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=\begin{cases} 50^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \\ 130^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \end{cases}\\ k=1&\: \: \textrm{tidak ada yang memenuhi}\\ \textrm{HP}&=\left \{ 50^{\circ},130^{\circ} \right \} \end{aligned}\\ &\color{black}\begin{aligned}.\: \quad\textrm{b}.\quad\cos x&=\cos 50^{\circ}\\ x&=\begin{cases} 50^{\circ} & +k.360^{\circ}\\ -50^{\circ} & +k.360^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=\begin{cases} 50^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \\ -50^{\circ}\: \: (\color{red}\textrm{tidak memenuhi}) & \end{cases}\\ k=1&\\ x&=\begin{cases} 50^{\circ}+360^{\circ}=410^{\circ}\: \: (\color{red}\textrm{tidak memenuhi}) & \\ -50^{\circ}+360^{\circ}=310^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \end{cases}\\ \textrm{HP}&=\left \{ 50^{\circ},310^{\circ} \right \} \end{aligned} \\ &\color{black}\begin{aligned}.\: \quad\textrm{c}.\quad\tan x&=\tan 50^{\circ}\\ x&=50^{\circ}+k.180^{\circ}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=50^{\circ}\: \: \color{magenta}\textrm{memenuhi}\\ k=1&\\ x&=50^{\circ}+180^{\circ}=230^{\circ}\: \: \color{magenta}\textrm{memenuhi}\\ k=2&\\ x&=50^{\circ}+360^{\circ}=410^{\circ}\: \: \color{red}\textrm{tidak memenuhi}\\ \textrm{HP}&=\left \{ 50^{\circ},230^{\circ} \right \} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 2.&\textrm{Tentukanlah himpunan penyelesaian dari }\\ &\textrm{persamaan-persamaan trigonometri berikut}\\ &\textrm{ini untuk}\: \: 0^{\circ}\leq x\leq 360^{\circ}\\ &\begin{array}{lllllll}\\ \textrm{a}.& \sin x=\displaystyle \frac{1}{2}&\textrm{f}.& \tan x=-\displaystyle \frac{1}{3}\sqrt{3}&\textrm{k}.& \sin 2x=\displaystyle \frac{1}{2}\\ \textrm{b}.& \cos x=\displaystyle \frac{1}{2}\sqrt{3}&\textrm{g}& 2\cos x=-\sqrt{3}&\textrm{l}.& \cos 2x=-\displaystyle \frac{1}{2}\sqrt{3}\\ \textrm{c}.& \tan x=\sqrt{3}&\textrm{h}& 3\tan x=\sqrt{3}&\textrm{m}.& \tan 2x=\sqrt{3}\\ \textrm{d}.& \sin x=-1&\textrm{i}.& \sin x=\sin 46^{\circ}&\textrm{n}.& \sin \left ( 2x-30^{\circ} \right )=\sin 45^{\circ}\\ \textrm{e}.& \cos x=-\displaystyle \frac{1}{2}\sqrt{2}&\textrm{j}.& \cos x=\cos 93^{\circ}&\textrm{o}.& \sin \left ( 2x+60^{\circ} \right )=\sin 90^{\circ}\\ \end{array}\\ \end{array}$

$.\: \quad\color{blue}\textrm{Jawab}:$

$\color{black}\begin{aligned}.\: \quad\textrm{a}.\quad\sin x&=\displaystyle \frac{1}{2}\\ \sin x&=\sin 30^{\circ}\\ x&=\begin{cases} 30^{\circ} & +k.360^{\circ}\\ \left (180^{\circ}-30^{\circ} \right ) & +k.360^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=\begin{cases} 30^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \\ 150^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \end{cases}\\ k=1&\: \: \textrm{tidak ada yang memenuhi}\\ \textrm{HP}&=\left \{ 30^{\circ},150^{\circ} \right \} \end{aligned}$

$\color{black}\begin{aligned}.\: \quad\textrm{b}.\quad\cos x&=\displaystyle \frac{1}{2}\sqrt{3}\\ \cos x&=\cos 30^{\circ}\\ x&=\begin{cases} 30^{\circ} & +k.360^{\circ}\\ -30^{\circ} & +k.360^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=\begin{cases} 30^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \\ -30^{\circ}\: \: (\color{red}\textrm{tidak memenuhi}) & \end{cases}\\ k=1&\\ x&=\begin{cases} 30^{\circ}+360^{\circ}=390^{\circ}\: \: (\color{red}\textrm{tidak memenuhi}) & \\ -30^{\circ}+360^{\circ}=330^{\circ}\: \: (\color{magenta}\textrm{memenuhi}) & \end{cases}\\ \textrm{HP}&=\left \{ 30^{\circ},330^{\circ} \right \} \end{aligned}$

$\color{purple}\begin{aligned}.\: \quad\textrm{c}.\quad\tan x&=\sqrt{3}\\ \tan x&=\tan 60^{\circ}\\ x&=60^{\circ}+k.180^{\circ}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=60^{\circ}\: \: \color{magenta}\textrm{memenuhi}\\ k=1&\\ x&=60^{\circ}+180^{\circ}=240^{\circ}\: \: \color{magenta}\textrm{memenuhi}\\ k=2&\\ x&=60^{\circ}+360^{\circ}=420^{\circ}\: \: \color{red}\textrm{tidak memenuhi}\\ \textrm{HP}&=\left \{ 60^{\circ},240^{\circ} \right \} \end{aligned}$

$\color{black}\begin{aligned}.\: \quad\textrm{d}.\quad\sin x&=-1\\ \sin x&= \sin 270^{\circ}\\ x&=\begin{cases} 270^{\circ} & +k.360^{\circ} \\ \left ( 180^{\circ}-270^{\circ} \right ) & +k.360^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}\\ x&=\begin{cases} 270^{\circ} & \color{magenta}\textrm{memenuhi} \\ -90^{\circ} & \color{red}\textrm{tidak memenuhi} \end{cases}\\ k=1&\: \: \textrm{tidak memenuhi semuanya}\\ \textrm{HP}&=\left \{ 270^{\circ} \right \} \end{aligned}$.

$\color{black}\begin{aligned}.\: \quad\textrm{k}.\quad\sin 2x&=\displaystyle \frac{1}{2}\\ \sin 2x&=\sin 30^{\circ}\\ 2x&=\begin{cases} 30^{\circ} & +k.360^{\circ}\\ \left (180^{\circ}-30^{\circ} \right ) & +k.360^{\circ} \end{cases}\\ \color{red}\textrm{sehin}&\color{red}\textrm{gga}\\ x&=\begin{cases} 15^{\circ} & +k.180^{\circ}\\ \left (90^{\circ}-15^{\circ} \right ) & +k.180^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=\begin{cases} 15^{\circ}\: \: (\color{blue}\textrm{memenuhi}) & \\ 75^{\circ}\: \: (\color{blue}\textrm{memenuhi}) & \end{cases}\\ k=1&\: \: \textrm{diperoleh}:\\ x&=\begin{cases} 15^{\circ}+180^{\circ}=195^{\circ}\: \: (\color{blue}\textrm{memenuhi}) & \\ 75^{\circ}+180^{\circ}=255^{\circ}\: \: (\color{blue}\textrm{memenuhi}) & \end{cases}\\ k=2&\: \: \textrm{tidak ada yang memenuhi}\\ \textrm{HP}&=\left \{ 15^{\circ},75^{\circ},195^{\circ},255^{\circ} \right \} \end{aligned}$.

$\color{black}\begin{aligned}.\: \quad\textrm{l}.\quad\cos 2x&=-\displaystyle \frac{1}{2}\sqrt{3}\\ \cos 2x&=-\cos 30^{\circ}=\cos \left ( 180^{\circ}-30^{\circ} \right )=\cos 150^{\circ}\\ 2x&=\begin{cases} 150^{\circ} & +k.360^{\circ}\\ -150^{\circ} & +k.360^{\circ} \end{cases}\\ \color{red}\textrm{sehin}&\color{red}\textrm{gga}\\ x&=\begin{cases} 75^{\circ} & +k.180^{\circ}\\ -75^{\circ} & +k.180^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=\begin{cases} 75^{\circ}\: \: (\color{blue}\textrm{memenuhi}) & \\ -75^{\circ}\: \: (\color{red}\textrm{tidak memenuhi}) & \end{cases}\\ k=1&\\ x&=\begin{cases} 75^{\circ}+180^{\circ}=255^{\circ}\: \: (\color{blue}\textrm{memenuhi}) & \\ -75^{\circ}+180^{\circ}=105^{\circ}\: \: (\color{blue}\textrm{memenuhi}) & \end{cases}\\ k=2&\\ x&=\begin{cases} 75^{\circ}+360^{\circ}=435^{\circ}\: \: (\color{red}\textrm{tidak memenuhi}) & \\ -75^{\circ}+360^{\circ}=285^{\circ}\: \: (\color{blue}\textrm{memenuhi}) & \end{cases}\\ k=3&\: \: \textrm{tidak ada yang memenuhi}\\ \textrm{HP}&=\left \{ 75^{\circ},105^{\circ},255^{\circ},285^{\circ} \right \} \end{aligned}$.

$\color{black}\begin{aligned}.\: \quad\textrm{m}.\quad\tan 2x&=\sqrt{3}\\ \tan 2x&=\tan 60^{\circ}\\ 2x&=60^{\circ}+k.180^{\circ}\\ \color{red}\textrm{sehin}&\color{red}\textrm{gga}\\ x&=30^{\circ}+k.90^{\circ}\\ k=0&\: \: \textrm{diperoleh}:\\ x&=30^{\circ}\: \: \color{blue}\textrm{memenuhi}\\ k=1&\\ x&=30^{\circ}+90^{\circ}=120^{\circ}\: \: \color{blue}\textrm{memenuhi}\\ k=2&\\ x&=30^{\circ}+180^{\circ}=210^{\circ}\: \: \color{blue}\textrm{memenuhi}\\ k=3&\\ x&=30^{\circ}+270^{\circ}=300^{\circ}\: \: \color{blue}\textrm{memenuhi}\\ k=4&\\ x&=30^{\circ}+360^{\circ}=390^{\circ}\: \: \color{red}\textrm{tidak memenuhi}\\ \textrm{HP}&=\left \{ 30^{\circ},120^{\circ},210^{\circ},300^{\circ} \right \} \end{aligned}$.

$\color{black}\begin{aligned}.\: \quad\textrm{n}.\quad\sin \left ( 2x-30^{\circ} \right )&=\sin 45^{\circ}\\ \left ( 2x-30^{\circ} \right )&=\begin{cases} 45^{\circ} & +k.360^{\circ} \\ \left ( 180^{\circ}-45^{\circ} \right ) & +k.360^{\circ} \end{cases}\\ 2x&=\begin{cases} 45^{\circ}+30^{\circ} &+k.360^{\circ} \\ 135^{\circ}+30^{\circ} &+ k.360^{\circ} \end{cases}\\ x&=\begin{cases} 37,5^{\circ} & +k.180^{\circ} \\ 82,5^{\circ} & +k.180^{\circ} \end{cases}\\ k=0&\: \: \textrm{diperoleh}\\ x&=\begin{cases} 37,5^{\circ} & \\ 82,5^{\circ} & \end{cases}\\ k=1&\: \: \textrm{diperoleh}\\ x&=\begin{cases} 37,5^{\circ}+180^{\circ} &=217,5^{\circ} \\ 82,5^{\circ}+180^{\circ} &=262,5^{\circ} \end{cases}\\ k=2&\: \: \color{red}\textrm{tidak ada yang memenuhi}\\ \textrm{HP}&=\left \{ 37,5^{\circ},82,5^{\circ},217,5^{\circ},262,5^{\circ} \right \} \end{aligned}$.

$\color{black}\begin{aligned}.\: \quad\textrm{o}.\quad\sin \left ( 2x+60^{\circ} \right )&=\sin 90^{\circ}\\ \left ( 2x+60^{\circ} \right )&=\begin{cases} 90^{\circ} & +k.360^{\circ} \\ \left ( 180^{\circ}-90^{\circ} \right ) & +k.360^{\circ} \end{cases}\\ 2x&=\begin{cases} 90^{\circ}-60^{\circ} &+k.360^{\circ} \\ 90^{\circ}-60^{\circ} &+ k.360^{\circ} \end{cases}\\ x&=15^{\circ}+k.180^{\circ}\\ k=0&\: \: \textrm{diperoleh}\\ x&=15^{\circ}\\ k=1&\: \: \textrm{diperoleh}\\ x&=15^{\circ}+180^{\circ}=195^{\circ}\\ k=2&\: \: \color{red}\textrm{tidak ada yang memenuhi}\\ \textrm{HP}&=\left \{ 15^{\circ},195^{\circ} \right \} \end{aligned}$