Relasi sudut

Relasi sudut

Relasi sudut

 

 

A.   Kuadran

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

 

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.


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