Relasi Sudut
1. Kuadran
2. Sudut Istimewa di kuadran I (\(0^\text{o} < \alpha < 90^\text{o}\))
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| Sin | 0 | \(\dfrac{1}{2}\) | \(\dfrac{1}{2}\sqrt{2}\) | \(\dfrac{1}{2}\sqrt{3}\) | 1 |
| Cos | 1 | \(\dfrac{1}{2}\sqrt{3}\) | \(\dfrac{1}{2}\sqrt{2}\) | \(\dfrac{1}{2}\) | 0 |
| Tan | 0 | \(\dfrac{1}{3} \sqrt{3}\) | 1 | \(\sqrt{3}\) | ∼ |
3. Sudut di kuadran II (\(90^\text{o} < \alpha < 180^\text{o}\))
| Menggunakan \((180^\text{o} - \alpha)\) | Menggunakan \((90^\text{o} + \alpha)\) |
|---|---|
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\begin{equation*} \begin{split} \sin 150^\text{o} & = \sin (180^\text{o} - 30^\text{o}) \\\\ \sin 150^\text{o} & = + \sin 30^\text{o} \\\\ \sin 150^\text{o} & = + \tfrac 12 \end{split} \end{equation*} 150º berada di kuadran II, maka sin 150º bernilai positif. |
\begin{equation*} \begin{split} \sin 150^\text{o} & = \sin (90^\text{o} + 60^\text{o}) \\\\ \sin 150^\text{o} & = + \cos 60^\text{o} \quad {\color {blue} \sin \rightarrow \cos}\\\\ \sin 150^\text{o} & = + \tfrac 12 \end{split} \end{equation*} 150º berada di kuadran II, maka sin 150º bernilai positif. |
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\begin{equation*} \begin{split} \cos 150^\text{o} & = \cos (180^\text{o} - 30^\text{o}) \\\\ \cos 150^\text{o} & = - \cos 30^\text{o} \\\\ \cos 150^\text{o} & = - \tfrac 12 \sqrt{3} \end{split} \end{equation*} 150º berada di kuadran II, maka cos 150º bernilai negatif. |
\begin{equation*} \begin{split} \cos 150^\text{o} & = \cos (90^\text{o} + 60^\text{o}) \\\\ \cos 150^\text{o} & = - \sin 60^\text{o} \quad {\color {blue} \cos \rightarrow \sin}\\\\ \cos 150^\text{o} & = - \tfrac 12 \sqrt{3} \end{split} \end{equation*} 150º berada di kuadran II, maka cos 150º bernilai negatif. |
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\begin{equation*} \begin{split} \tan 150^\text{o} & = \tan (180^\text{o} - 30^\text{o}) \\\\ \tan 150^\text{o} & = - \tan 30^\text{o} \\\\ \tan 150^\text{o} & = - \tfrac 13 \sqrt{3} \end{split} \end{equation*} 150º berada di kuadran II, maka tan 150º bernilai negatif. |
\begin{equation*} \begin{split} \tan 150^\text{o} & = \tan (90^\text{o} + 60^\text{o}) \\\\ \tan 150^\text{o} & = - \cot 60^\text{o} \quad {\color {blue} \tan \rightarrow \cot}\\\\ \tan 150^\text{o} & = - \tfrac 13 \sqrt{3} \end{split} \end{equation*} 150º berada di kuadran II, maka tan 150º bernilai negatif. |
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\begin{equation*} \begin{split} \sin 180^\text{o} & = \sin (180^\text{o} - 0^\text{o}) \\\\ \sin 180^\text{o} & = \sin 0^\text{o} \\\\ \sin 180^\text{o} & = 0 \end{split} \end{equation*} |
\begin{equation*} \begin{split} \sin 180^\text{o} & = \sin (90^\text{o} + 90^\text{o}) \\\\ \sin 180^\text{o} & = \cos 90^\text{o} \quad {\color {blue} \sin \rightarrow \cos}\\\\ \sin 180^\text{o} & = 0 \end{split} \end{equation*} |
4. Sudut di kuadran III (\(180^\text{o} < \alpha < 270^\text{o}\))
| Menggunakan \((180^\text{o} + \alpha)\) | Menggunakan \((270^\text{o} - \alpha)\) |
|---|---|
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\begin{equation*} \begin{split} \sin 225^\text{o} & = \sin (180^\text{o} + 45^\text{o}) \\\\ \sin 225^\text{o} & = − \sin 45^\text{o} \\\\ \sin 225^\text{o} & = − \tfrac 12 \sqrt{2} \end{split} \end{equation*} 225º berada di kuadran III, maka sin 225º bernilai negatif. |
\begin{equation*} \begin{split} \sin 225^\text{o} & = \sin (270^\text{o} - 45^\text{o}) \\\\ \sin 225^\text{o} & = − \cos 45^\text{o} \quad {\color {blue} \sin \rightarrow \cos}\\\\ \sin 225^\text{o} & = − \tfrac 12 \sqrt{2} \end{split} \end{equation*} 225º berada di kuadran III, maka sin 225º bernilai negatif. |
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\begin{equation*} \begin{split} \cos 225^\text{o} & = \cos (180^\text{o} + 45^\text{o}) \\\\ \cos 225^\text{o} & = - \cos 45^\text{o} \\\\ \cos 225^\text{o} & = - \tfrac 12 \sqrt{2} \end{split} \end{equation*} 225º berada di kuadran III, maka cos 225º bernilai negatif. |
\begin{equation*} \begin{split} \cos 225^\text{o} & = \cos (270^\text{o} - 45^\text{o}) \\\\ \cos 225^\text{o} & = - \sin 45^\text{o} \quad {\color {blue} \cos \rightarrow \sin}\\\\ \cos 225^\text{o} & = - \tfrac 12 \sqrt{2} \end{split} \end{equation*} 225º berada di kuadran III, maka cos 225º bernilai negatif. |
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\begin{equation*} \begin{split} \tan 225^\text{o} & = \tan (180^\text{o} + 45^\text{o}) \\\\ \tan 225^\text{o} & = + \tan 45^\text{o} \\\\ \tan 225^\text{o} & = + 1 \end{split} \end{equation*} 225º berada di kuadran III, maka tan 225º bernilai positif. |
\begin{equation*} \begin{split} \tan 225^\text{o} & = \tan (270^\text{o} - 45^\text{o}) \\\\ \tan 225^\text{o} & = + \cot 45^\text{o} \quad {\color {blue} \tan \rightarrow \cot}\\\\ \tan 225^\text{o} & = + 1 \end{split} \end{equation*} 225º berada di kuadran III, maka tan 225º bernilai positif. |
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\begin{equation*} \begin{split} \sin 270^\text{o} & = \sin (180^\text{o} + 90^\text{o}) \\\\ \sin 270^\text{o} & = -\sin 90^\text{o} \\\\ \sin 270^\text{o} & = -1 \end{split} \end{equation*} 270º berada di kuadran III atau IV, maka sin 270º bernilai negatif. |
\begin{equation*} \begin{split} \sin 270^\text{o} & = \sin (270^\text{o} - 0^\text{o}) \\\\ \sin 270^\text{o} & = -\cos 0^\text{o} \\\\ \sin 270^\text{o} & = -1 \end{split} \end{equation*}270º berada di kuadran III atau IV, maka sin 270º bernilai negatif. |
5. Sudut di kuadran IV (\(270^\text{o} < \alpha < 360^\text{o}\))
| Menggunakan \((360^\text{o} - \alpha)\) | Menggunakan \((270^\text{o} + \alpha)\) |
|---|---|
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\begin{equation*} \begin{split} \sin 300^\text{o} & = \sin (360^\text{o} - 60^\text{o}) \\\\ \sin 300^\text{o} & = - \sin 60^\text{o} \\\\ \sin 300^\text{o} & = - \tfrac 12 \sqrt{3} \end{split} \end{equation*} 300º berada di kuadran IV, maka sin 300º bernilai negatif. |
\begin{equation*} \begin{split} \sin 300^\text{o} & = \sin (270^\text{o} + 30^\text{o}) \\\\ \sin 300^\text{o} & = - \cos 30^\text{o} \quad {\color {blue} \sin \rightarrow \cos}\\\\ \sin 300^\text{o} & = - \tfrac 12 \sqrt{3} \end{split} \end{equation*} 300º berada di kuadran II, maka sin 300º bernilai positif. |
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\begin{equation*} \begin{split} \cos 300^\text{o} & = \cos (360^\text{o} - 60^\text{o}) \\\\ \cos 300^\text{o} & = + \cos 60^\text{o} \\\\ \cos 300^\text{o} & = + \tfrac 12 \end{split} \end{equation*} 300º berada di kuadran IV, maka cos 300º bernilai positif. |
\begin{equation*} \begin{split} \cos 300^\text{o} & = \cos (270^\text{o} + 30^\text{o}) \\\\ \cos 300^\text{o} & = + \sin 30^\text{o} \quad {\color {blue} \cos \rightarrow \sin}\\\\ \cos 300^\text{o} & = + \tfrac 12 \end{split} \end{equation*} 300º berada di kuadran II, maka cos 300º bernilai negatif. |
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\begin{equation*} \begin{split} \tan 300^\text{o} & = \tan (360^\text{o} - 60^\text{o}) \\\\ \tan 300^\text{o} & = - \tan 60^\text{o} \\\\ \tan 300^\text{o} & = - \sqrt{3} \end{split} \end{equation*} 300º berada di kuadran IV, maka tan 300º bernilai negatif. |
\begin{equation*} \begin{split} \tan 300^\text{o} & = \tan (270^\text{o} + 30^\text{o}) \\\\ \tan 300^\text{o} & = - \cot 30^\text{o} \quad {\color {blue} \tan \rightarrow \cot}\\\\ \tan 300^\text{o} & = - \sqrt{3} \end{split} \end{equation*} 300º berada di kuadran II, maka tan 300º bernilai negatif. |
Sudut-sudut Besar
Untuk sudut-sudut yang lebih besar dari 360°, yang dihitung adalah sisa sudut setelah dikurangi kelipatan 360°.
Contoh 01
\begin{equation*} \begin{split} \sin 1470^{\text{o}} & = \sin (360^{\text{o}} \:.\: 4 + 30^{\text{o}}) \\\\ \sin 1470^{\text{o}} & = \sin 30^{\text{o}} \\\\ \sin 1470^{\text{o}} & = \frac{1}{2} \end{split} \end{equation*}
Contoh 02
\begin{equation*} \begin{split} \tan 855^{\text{o}} & = \tan (360^{\text{o}} \:.\: 2 + 135^{\text{o}}) \\\\ \tan 855^{\text{o}} & = \tan 135^{\text{o}} \\\\ \tan 855^{\text{o}} & = \tan (180^{\text{o}} - 45^{\text{o}}) \\\\ \tan 855^{\text{o}} & = -\tan 45^{\text{o}} \\\\ \tan 855^{\text{o}} & = -1 \end{split} \end{equation*}
Sudut-sudut Negatif
Sudut negatif diasumsikan pada kuadran 4.
\(\sin (- \alpha) = - \sin \alpha\)
Sinus sudut pada kuadran 4 bernilai negatif.
\(\cos (- \alpha) = \cos \alpha \)
Cosinus sudut pada kuadran 4 bernilai positif.
\(\tan (- \alpha) = - \tan \alpha \)
Tangen sudut pada kuadran 4 bernilai negatif.
Contoh 01
\begin{equation*} \begin{split} \sin (-240^{\text{o}}) & = -\sin 240^{\text{o}} \\\\ \sin (-240^{\text{o}}) & = - \sin (180^{\text{o}} + 60^{\text{o}}) \\\\ \sin (-240^{\text{o}}) & = - (-\sin 60^{\text{o}}) \\\\ \sin (-240^{\text{o}}) & = \sin 60^{\text{o}} \\\\ \sin (-240^{\text{o}}) & = \frac{1}{2} \sqrt{3} \end{split} \end{equation*}
Contoh 02
\begin{equation*} \begin{split} \cos (-135^{\text{o}}) & = +\cos 135^{\text{o}} \\\\ \cos (-135^{\text{o}}) & = \cos (180^{\text{o}} - 45^{\text{o}}) \\\\ \cos (-135^{\text{o}}) & = -\cos 45^{\text{o}} \\\\ \cos (-135^{\text{o}}) & = -\frac{1}{2} \sqrt{2} \end{split} \end{equation*}
Sudut-sudut Sembarang
Sudut-sudut yang bukan sudut istimewa dicari hubungannya dengan menggunakan relasi sudut.
Contoh 01
Tentukan nilai eksak dari \(\dfrac{\sin 20^{\text{o}}}{\cos 110^{\text{o}}}\)
\begin{equation*} \begin{split} & \frac{\sin 20^{\text{o}}}{\cos 110^{\text{o}}} \\\\ & \frac{\sin 20^{\text{o}}}{\cos (90^{\text{o}} + 20^{\text{o}})} \\\\ & \frac{\sin 20^{\text{o}}}{- \sin 20^{\text{o}}} \\\\ & -1 \end{split} \end{equation*}
Contoh 02
Tentukan nilai eksak dari \( \tan 156^{\text{o}} \;.\: \dfrac{\cos 336^{\text{o}}}{\sin 204^{\text{o}}}\)
\begin{equation*} \begin{split} & \tan 156^{\text{o}} \;.\: \frac{\cos 336^{\text{o}}}{\sin 204^{\text{o}}} \\\\ & \frac {\sin 156^{\text{o}}}{\cos 156^{\text{o}}} \;.\: \frac{\cos 336^{\text{o}}}{\sin 204^{\text{o}}} \\\\ & \frac {\sin (180^{\text{o}} - 24^{\text{o}})}{\cos (180^{\text{o}} - 24^{\text{o}})} \:.\: \frac{\cos (360^{\text{o}} - 24^{\text{o}})}{\sin (180^{\text{o}} + 24^{\text{o}})} \\\\ & \frac {\sin 24^{\text{o}}}{-\cos 24^{\text{o}}} \;.\: \frac{\cos 24^{\text{o}}}{- \sin 24^{\text{o}}} \\\\ & 1 \end{split} \end{equation*}
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