Letak Kuadran
- Pada kuadran I \([0^{\circ} – 90^{\circ}]\) semua fungsi trigonometri bernilai positif
- Pada kuadran II \([90^{\circ} – 180^{\circ}]\) fungsi sinus dan juga cosecan bernilai positif, fungsi trigonometri yang lainnya bernilai negatif
- Pada kuadran III \([180^{\circ} – 270^{\circ}]\) fungsi tangen dan juga cotangen bernilai positif, fungsi trigonometri yang lainnya bernilai negatif
- Pada kuadran IV \([270^{\circ} – 360^{\circ}]\) fungsi cosinus dan juga secan bernilai postif, fungsi trigonometri yang lainnya bernilai negatif
Relasi Sudut \((90^{\circ} -\: \alpha)\) dan \((90^{\circ} + \alpha)\)
\(\text{Fungsi trigonometri berubah}\)
| \((90^{\circ} -\: \alpha)\) | \((90^{\circ} + \alpha)\) |
| $$\sin (90^{\circ} -\:\alpha) = \cos \alpha$$ | $$\sin (90^{\circ} +\:\alpha) =\: \cos \alpha$$ |
| $$\cos (90^{\circ} -\:\alpha) = \sin \alpha$$ | $$\cos (90^{\circ} +\:\alpha) = -\sin \alpha$$ |
| $$\tan (90^{\circ} -\:\alpha) = \cot \alpha$$ | $$\tan (90^{\circ} +\:\alpha) = -\cot \alpha$$ |
| $$\csc (90^{\circ} -\:\alpha) = \sec \alpha$$ | $$\csc (90^{\circ} +\:\alpha) = \:\sec \alpha$$ |
| $$\sec (90^{\circ} -\:\alpha) = \csc \alpha$$ | $$\sec (90^{\circ} +\:\alpha) = -\csc \alpha$$ |
| $$\cot (90^{\circ} -\:\alpha) = \tan \alpha$$ | $$\cot (90^{\circ} +\:\alpha) = -\tan \alpha$$ |
- \(\sin{30^{\circ}} = \sin{(90^{\circ} – 60^{\circ})} = +\cos 60^{\circ} = +\dfrac{1}{2}\)
\(\sin{30^{\circ}}\) berada di kuadran I sehingga nilainya positif
- \(\cos{45^{\circ}} = \cos{(90^{\circ} – 45^{\circ})} = +\sin 45^{\circ} = +\dfrac{1}{2}\sqrt{2}\)
\(\cos{45^{\circ}}\) berada di kuadran I sehingga nilainya positif
- \(\tan{60^{\circ}} = \tan{(90^{\circ} – 30^{\circ})} = +\cot 30^{\circ} = +\sqrt{3}\)
\(\tan{60^{\circ}}\) berada di kuadran I sehingga nilainya positif
- \(\sin{120^{\circ}} = \sin{(90^{\circ} + 30^{\circ})} = +\cos 30^{\circ} = +\dfrac{1}{2}\sqrt{3}\)
\(\sin{120^{\circ}}\) berada di kuadran II sehingga nilainya positif
- \(\cos{150^{\circ}} = \cos{(90^{\circ} + 60^{\circ})} = -\sin 60^{\circ} = -\dfrac{1}{2}\sqrt{3}\)
\(\cos{150^{\circ}}\) berada di kuadran II sehingga nilainya negatif
- \(\tan{135^{\circ}} = \tan{(90^{\circ} + 45^{\circ})} = -\cot 45^{\circ} = -1\)
\(\tan{135^{\circ}}\) berada di kuadran II sehingga nilainya negatif
Relasi Sudut \((180^{\circ} -\: \alpha)\) dan \((180^{\circ} + \alpha)\)
\(\text{Fungsi trigonometri tetap}\)
| \((180^{\circ} -\: \alpha)\) | \((180^{\circ} + \alpha)\) |
| $$\sin (180^{\circ} -\:\alpha) = \:\sin\alpha$$ | $$\sin (180^{\circ} +\:\alpha) =-\sin\alpha$$ |
| $$\cos (180^{\circ} -\:\alpha) = -\cos \alpha$$ | $$\cos (180^{\circ} +\:\alpha) = -\cos \alpha$$ |
| $$\tan (180^{\circ} -\:\alpha) = -\tan\alpha$$ | $$\tan (180^{\circ} +\:\alpha) = \:\tan \alpha$$ |
| $$\csc (180^{\circ} -\:\alpha) = \:\csc \alpha$$ | $$\csc (180^{\circ} +\:\alpha) = -\csc \alpha$$ |
| $$\sec (180^{\circ} -\:\alpha) = -\sec \alpha$$ | $$\sec (180^{\circ} +\:\alpha) = -\sec \alpha$$ |
| $$\cot (180^{\circ} -\:\alpha) = -\cot\alpha$$ | $$\cot (180^{\circ} +\:\alpha) = \:\cot \alpha$$ |
- \(\sin{135^{\circ}} = \sin{(180^{\circ} – 45^{\circ})} = +\sin 45^{\circ} = +\dfrac{1}{2}\sqrt{2}\)
\(\sin{135^{\circ}}\) berada di kuadran II sehingga nilainya positif
- \(\sec{150^{\circ}} = \sec{(180^{\circ} – 30^{\circ})} = -\sec 30^{\circ} = -\dfrac{2}{3}\sqrt{3}\)
\(\sec{150^{\circ}}\) berada di kuadran II sehingga nilainya negatif
- \(\csc{120^{\circ}} = \csc{(180^{\circ} – 60^{\circ})} = +\csc 60^{\circ} = +\dfrac{2}{3}\sqrt{3}\)
\(\csc{120^{\circ}}\) berada di kuadran II sehingga nilainya positif
- \(\tan{225^{\circ}} = \tan{(180^{\circ} + 45^{\circ})} = +\tan 45^{\circ} = +1\)
\(\tan{225^{\circ}}\) berada di kuadran III sehingga nilainya positif
- \(\cos{210^{\circ}} = \cos{(180^{\circ} + 30^{\circ})} = -\cos 30^{\circ} = -\dfrac{1}{2}\sqrt{3}\)
\(\cos{210^{\circ}}\) berada di kuadran III sehingga nilainya negatif
- \(\sin{240^{\circ}} = \sin{(180^{\circ} + 60^{\circ})} = -\sin 60^{\circ} =-\dfrac{1}{2}\sqrt{3}\)
\(\sin{240^{\circ}}\) berada di kuadran III sehingga nilainya negatif
Relasi Sudut \((270^{\circ} -\: \alpha)\) dan \((270^{\circ} + \alpha)\)
\(\text{Fungsi trigonometri berubah}\)
| \((270^{\circ} -\: \alpha)\) | \((270^{\circ} + \alpha)\) |
| $$\sin (270^{\circ} -\:\alpha) = -\cos\alpha$$ | $$\sin (270^{\circ} +\:\alpha) =-\cos\alpha$$ |
| $$\cos (270^{\circ} -\:\alpha) = -\sin \alpha$$ | $$\cos (270^{\circ} +\:\alpha) = \:\sin\alpha$$ |
| $$\tan (270^{\circ} -\:\alpha) = \:\cot\alpha$$ | $$\tan (270^{\circ} +\:\alpha) = -\cot \alpha$$ |
| $$\csc (270^{\circ} -\:\alpha) = -\sec \alpha$$ | $$\csc (270^{\circ} +\:\alpha) = -\sec \alpha$$ |
| $$\sec (270^{\circ} -\:\alpha) = -\csc \alpha$$ | $$\sec (270^{\circ} +\:\alpha) = \:\csc\alpha$$ |
| $$\cot (270^{\circ} -\:\alpha) = \:\tan\alpha$$ | $$\cot (270^{\circ} +\:\alpha) = -\tan\alpha$$ |
- \(\sin{240^{\circ}} = \sin{(270^{\circ} – 30^{\circ})} = -\cos 30^{\circ} = -\dfrac{1}{2}\sqrt{3}\)
\(\sin{240^{\circ}}\) berada di kuadran III sehingga nilainya negatif
- \(\cos{240^{\circ}} = \cos{(270^{\circ} – 30^{\circ})} = -\sin 30^{\circ} = -\dfrac{1}{2}\)
\(\cos{240^{\circ}}\) berada di kuadran III sehingga nilainya negatif
- \(\sec{225^{\circ}} = \sec{(270^{\circ} – 45^{\circ})} = -\csc 45^{\circ} = -\sqrt{2}\)
\(\sec{225^{\circ}}\) berada di kuadran III sehingga nilainya negatif
- \(\tan{300^{\circ}} = \tan{(270^{\circ} + 30^{\circ})} = -\cot 30^{\circ} = -\sqrt{3}\)
\(\tan{300^{\circ}}\) berada di kuadran IV sehingga nilainya negatif
- \(\csc{315^{\circ}} = \csc{(270^{\circ} + 45^{\circ})} = -\sec 45^{\circ} = -\sqrt{2}\)
\(\csc{315^{\circ}}\) berada di kuadran IV sehingga nilainya negatif
- \(\cos{330^{\circ}} = \cos{(270^{\circ} + 60^{\circ})} = +\sin 60^{\circ} = +\dfrac{1}{2}\sqrt{3}\)
\(\cos{330^{\circ}}\) berada di kuadran IV sehingga nilainya positif
Relasi Sudut \((360^{\circ} -\: \alpha)\)
\(\text{Fungsi trigonometri tetap}\)
| \((360^{\circ} -\: \alpha)\) |
| $$\sin (360^{\circ} -\:\alpha) = -\sin\alpha$$ |
| $$\cos (360^{\circ} -\:\alpha) = \:\cos \alpha$$ |
| $$\tan (360^{\circ} -\:\alpha) = -\tan\alpha$$ |
| $$\csc (360^{\circ} -\:\alpha) = -\csc\alpha$$ |
| $$\sec (360^{\circ} -\:\alpha) = \:\sec \alpha$$ |
| $$\cot (360^{\circ} -\:\alpha) = -\cot\alpha$$ |
- \(\sec{300^{\circ}} = \sec{(360^{\circ} – 60^{\circ})} = +\sec 60^{\circ} = +2\)
\(\sec{300^{\circ}}\) berada di kuadran IV sehingga nilainya positif
- \(\sin{330^{\circ}} = \sin{(360^{\circ} – 30^{\circ})} = -\sin 30^{\circ} = -\dfrac{1}{2}\)
\(\sin{330^{\circ}}\) berada di kuadran IV sehingga nilainya negatif
- \(\tan{315^{\circ}} = \tan{(360^{\circ} – 45^{\circ})} = -\tan 45^{\circ} = -1\)
\(\tan{315^{\circ}}\) berada di kuadran IV sehingga nilainya negatif
