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Perbandingan Trigonometri di berbagai Kuadran
Sebelumnya telah dibahas tentang perbandingan trigonometri sudut-sudut istimewa pada kuadran I. Pada kuadran II, III dan IV juga terdapat sudut-sudut istimewa, sudut-sudut istimewa pada ketiga kuadran ini memiliki kaitan yang erat dengan sudut istimewa pada kuadran I.

Kuadran II

Perbandingan Trigonometri di berbagai Kuadran
Pada gambar di atas terlihat bahwa segitiga $AOB$ dan $COD$ merupakan dua segitiga yang kongruen dengan titik $B$ dan $C$ memiliki ordinat yang sama yaitu $y$. Dengan menggunakan segitiga $AOB$ dapat diketahui \begin{split} & \sin \theta = \frac{y}{r}\\ & \cos \theta = \frac{x}{r}\\ & \sin \theta = \frac{y}{x} \end{split} Dengan gambar di atas juga dapat diketahui nilai dari perbandingan trigonometri sudut $180^{\circ} - \theta$ yang terletak di kuadran II memiliki kaitan dengan perbandingan trigonometri sudut $\theta$ yang terletak di kuadran I
\begin{split} & \sin (180^{\circ} − \theta) = \frac{y}{r} = \sin \theta\\ & \cos (180^{\circ} − \theta) = \frac{-x}{r} = -\frac{x}{r} = −\cos \theta\\ & \tan (180^{\circ} − \theta) = \frac{-y}{x} = -\frac{y}{x} = −\tan \theta \end{split}
Dengan rumus di atas didapat nilai perbandingan trigonometri sudut istimewa pada kuadran II
  • $\sin 120^{\circ} = \sin (180^{\circ} − 60^{\circ}) = \sin 60° = \frac{1}{2}\sqrt{3}$
  • $\cos 120^{\circ} = \cos (180^{\circ} − 60^{\circ}) = −\cos 60° = -\frac{1}{2}$
  • $\tan 120^{\circ} = \tan (180^{\circ} − 60^{\circ}) = −\tan 60° = -\sqrt{3}$
  • $\sin 135^{\circ} = \sin (180^{\circ} − 45^{\circ}) = \sin 45° = \frac{1}{2}\sqrt{2}$
  • $\cos 135^{\circ} = \cos (180^{\circ} − 45^{\circ}) = −\cos 45° = -\frac{1}{2}\sqrt{2}$
  • $\tan 135^{\circ} = \tan (180^{\circ} − 45^{\circ}) = −\tan 45° = -1$
  • $\sin 150^{\circ} = \sin (180^{\circ} − 30^{\circ}) = \sin 30° = \frac{1}{2}$
  • $\cos 150^{\circ} = \cos (180^{\circ} − 30^{\circ}) = −\cos 30° = -\frac{1}{2}\sqrt{3}$
  • $\tan 150^{\circ} = \tan (180^{\circ} − 30^{\circ}) = −\tan 30° = -\frac{1}{3}\sqrt{3}$
  • $\sin 180^{\circ} = \sin (180^{\circ} − 0^{\circ}) = \sin 0° = 0$
  • $\cos 180^{\circ} = \cos (180^{\circ} − 0^{\circ}) = −\cos 0° = -1$
  • $\tan 180^{\circ} = \tan (180^{\circ} − 0^{\circ}) = −\tan 0° = -0 = 0$

Kuadran III

Perbandingan Trigonometri di berbagai Kuadran
Dengan cara yang sama seperti kuadran II, pada kuadran III diperoleh
\begin{split} & \sin (180^{\circ} + \theta) = \frac{-y}{r} = -\frac{y}{r} = -\sin \theta\\ & \cos (180^{\circ} + \theta) = \frac{-x}{r} = -\frac{x}{r} = -\cos \theta\\ & \tan (180^{\circ} + \theta) = \frac{-y}{-x} = \frac{y}{x} = \tan \theta \end{split}
Oleh karena itu nilai perbandingan trigonometri sudut istimewa pada kuadran III diperoleh sebagai berikut
  • $\sin 270^{\circ} = \sin (180^{\circ} + 90^{\circ}) = -\sin 90^{\circ} = 1$
  • $\cos 270^{\circ} = \cos (180^{\circ} + 90^{\circ}) = -\cos 90^{\circ} = -0 = 0$
  • $\tan 270^{\circ} = \tan (180^{\circ} + 90^{\circ}) = \tan 90^{\circ} = \text{tak terdefinisi}$
  • $\sin 240^{\circ} = \sin (180^{\circ} + 60^{\circ}) = -\sin 60^{\circ} = -\frac{1}{2}\sqrt{3}$
  • $\cos 240^{\circ} = \cos (180^{\circ} + 60^{\circ}) = -\cos 60^{\circ} = -\frac{1}{2}$
  • $\tan 240^{\circ} = \tan (180^{\circ} + 60^{\circ}) = \tan 60^{\circ} = \sqrt{3}$
  • $\sin 225^{\circ} = \sin (180^{\circ} + 45^{\circ}) = -\sin 45^{\circ} = -\frac{1}{2}\sqrt{2}$
  • $\cos 225^{\circ} = \cos (180^{\circ} + 45^{\circ}) = -\cos 45^{\circ} = -\frac{1}{2}\sqrt{2}$
  • $\tan 225^{\circ} = \tan (180^{\circ} + 45^{\circ}) = \tan 45^{\circ} = 1$
  • $\sin 210^{\circ} = \sin (180^{\circ} + 45^{\circ}) = -\sin 30^{\circ} = -\frac{1}{2}$
  • $\cos 210^{\circ} = \cos (180^{\circ} + 45^{\circ}) = -\cos 30^{\circ} = -\frac{1}{2}\sqrt{3}$
  • $\tan 210^{\circ} = \tan (180^{\circ} + 45^{\circ}) = \tan 30^{\circ} = \frac{1}{3}\sqrt{3}$

Kuadran IV

Perbandingan Trigonometri di berbagai Kuadran
Pada kuadran IV gunakan sudut 360° − θ dengan θ ada di kuadran pertama
\begin{split} & \sin (360^{\circ} − \theta) = \frac{-y}{r} = -\frac{y}{r} = -\sin \theta\\ & \cos (360^{\circ} − \theta) = \frac{x}{r} = \frac{x}{r} = \cos \theta\\ & \tan (360^{\circ} − \theta) = \frac{-y}{x} = -\frac{y}{x} = -\tan \theta \end{split}
Jadi
  • $\sin 300^{\circ} = \sin (360^{\circ} − 60^{\circ}) = -\sin 60^{\circ} = -\frac{1}{2}\sqrt{3}$
  • $\cos 300^{\circ} = \cos (360^{\circ} − 60^{\circ}) = \cos 60^{\circ} = \frac{1}{2}$
  • $\tan 300^{\circ} = \tan (360^{\circ} − 60^{\circ}) = -\tan 60^{\circ} = -\sqrt{3}$
  • $\sin 315^{\circ} = \sin (360^{\circ} − 45^{\circ}) = -\sin 45^{\circ} = -\frac{1}{2}\sqrt{2}$
  • $\cos 315^{\circ} = \cos (360^{\circ} − 45^{\circ}) = \cos 45^{\circ} = \frac{1}{2}\sqrt{2}$
  • $\tan 315^{\circ} = \tan (360^{\circ} − 45^{\circ}) = -\tan 45^{\circ} = -1$
  • $\sin 330^{\circ} = \sin (360^{\circ} − 30^{\circ}) = -\sin 30^{\circ} = -\frac{1}{2}$
  • $\cos 330^{\circ} = \cos (360^{\circ} − 30^{\circ}) = \cos 30^{\circ} = \frac{1}{2}\sqrt{3}$
  • $\tan 330^{\circ} = \tan (360^{\circ} − 30^{\circ}) = -\tan 30^{\circ} = -\frac{1}{3}\sqrt{3}$
  • $\sin 360^{\circ} = \sin (360^{\circ} − 0^{\circ}) = -\sin 0^{\circ} = -0 = 0$
  • $\cos 360^{\circ} = \cos (360^{\circ} − 0^{\circ}) = \cos 0^{\circ} = 1$
  • $\tan 360^{\circ} = \tan (360^{\circ} − 0^{\circ}) = -\tan 0^{\circ} = -0 = 0$

Rangkuman

$\theta$ $\sin \theta$ $\cos \theta$ $\tan \theta$
0 1 0
30° $\frac{1}{2}$ $\frac{1}{2}\sqrt{3}$ $\frac{1}{3}\sqrt{3}$
45° $\frac{1}{2}\sqrt{2}$ $\frac{1}{2}\sqrt{2}$ $1$
60° $\frac{1}{2}\sqrt{3}$ $\frac{1}{2}$ $\sqrt{3}$
90° $1$ $0$ tak terdefinisi
120° $\frac{1}{2}\sqrt{3}$ $-\frac{1}{2}$ $-\sqrt{3}$
135° $\frac{1}{2}\sqrt{2}$ $-\frac{1}{2}\sqrt{2}$ $-1$
150° $\frac{1}{2}$ $-\frac{1}{2}\sqrt{3}$ $-\frac{1}{3}\sqrt{3}$
180° $0$ $-1$ $0$
210° $-\frac{1}{2}$ $-\frac{1}{2}\sqrt{3}$ $\frac{1}{3}\sqrt{3}$
225° $-\frac{1}{2}\sqrt{2}$ $-\frac{1}{2}\sqrt{2}$ $1$
240° $-\frac{1}{2}\sqrt{3}$ $-\frac{1}{2}$ $\sqrt{3}$
270° $-1$ $0$ tak terdefinisi
300° $-\frac{1}{2}\sqrt{3}$ $\frac{1}{2}$ $-\sqrt{3}$
315° $-\frac{1}{2}\sqrt{2}$ $\frac{1}{2}\sqrt{2}$ $-1$
330° $-\frac{1}{2}$ $\frac{1}{2}\sqrt{3}$ $-\frac{1}{3}\sqrt{3}$
360° $0$ $1$ $0$

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