# Relasi sudut

## Konsep Dasar

#### B.   Sudut Istimewa di kuadran I ($$0^\text{o} < \alpha < 90^\text{o}$$)

 0° 30° 45° 60° 90° Sin 0 $$\frac{1}{2}$$ $$\frac{1}{2}\sqrt{2}$$ $$\frac{1}{2}\sqrt{3}$$ 1 Cos 1 $$\frac{1}{2}\sqrt{3}$$ $$\frac{1}{2}\sqrt{2}$$ $$\frac{1}{2}$$ 0 Tan 0 $$\frac{1}{3} \sqrt{3}$$ 1 $$\sqrt{3}$$ ∼

#### C.   Sudut di kuadran II ($$90^\text{o} < \alpha < 180^\text{o}$$)

 Menggunakan $$(180^\text{o} - \alpha)$$ Menggunakan $$(90^\text{o} + \alpha)$$ \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. \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. \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. \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*}

#### D.   Sudut di kuadran III ($$180^\text{o} < \alpha < 270^\text{o}$$)

 Menggunakan $$(180^\text{o} + \alpha)$$ Menggunakan $$(270^\text{o} - \alpha)$$ \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. \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. \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. \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.

#### E.   Sudut di kuadran IV ($$270^\text{o} < \alpha < 360^\text{o}$$)

 Menggunakan $$(360^\text{o} - \alpha)$$ Menggunakan $$(270^\text{o} + \alpha)$$ \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. \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. \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.