Jika a merupakan sudut di kuadran III dengan sin

sin(180-a) = -1/3, cos(180-a) = 2√2/3, tan(180-a) = -√2/4.

Pembahasan

Diketahui \(a\) adalah sudut di kuadran III dengan \(\sin a = -1/3\).

Kita perlu menentukan nilai \(\sin(180^\circ - a)\), \(\cos(180^\circ - a)\), dan \(\tan(180^\circ - a)\).

Karena \(a\) berada di kuadran III, maka \(\cos a\) bernilai negatif dan \(\tan a\) bernilai positif.
Kita dapat mencari nilai \(\cos a\) menggunakan identitas \(\sin^2 a + \cos^2 a = 1\):
\((-1/3)^2 + \cos^2 a = 1\)
\(1/9 + \cos^2 a = 1\)
\(\cos^2 a = 1 - 1/9\)
\(\cos^2 a = 8/9\)
\(\cos a = \pm \sqrt{8/9} = \pm 2\sqrt{2}/3\)
Karena \(a\) di kuadran III, \(\cos a = -2\sqrt{2}/3\).

Selanjutnya, kita cari \(\tan a\):
\(\tan a = \sin a / \cos a = (-1/3) / (-2\sqrt{2}/3) = 1/(2\sqrt{2}) = \sqrt{2}/4\).

Sekarang kita gunakan rumus sudut berelasi:
1. \(\sin(180^\circ - a)\):
   Dalam sudut berelasi, \(\sin(180^\circ - a) = \sin a\). 
   Jadi, \(\sin(180^\circ - a) = -1/3\).

2. \(\cos(180^\circ - a)\):
   Dalam sudut berelasi, \(\cos(180^\circ - a) = -\cos a\).
   Karena \(\cos a = -2\sqrt{2}/3\), maka \(\cos(180^\circ - a) = -(-2\sqrt{2}/3) = 2\sqrt{2}/3\).

3. \(\tan(180^\circ - a)\):
   Dalam sudut berelasi, \(\tan(180^\circ - a) = -\tan a\).
   Karena \(\tan a = \sqrt{2}/4\), maka \(\tan(180^\circ - a) = -\sqrt{2}/4\).

Jadi, \(\sin(180^\circ - a) = -1/3\), \(\cos(180^\circ - a) = 2\sqrt{2}/3\), dan \(\tan(180^\circ - a) = -\sqrt{2}/4\).