1^ste manier :
cos 2x + cos x = 0
cos 2x = − cos x
cos 2x = cos (180−x)
2x = 180°−x + k.360°   ∨   2x = −180°+x + k.360°
3x = 180° + k.360°   ∨   x = –180° + k.360°
x = 60° + k.360°   ∨   x = –180° + k.360°
x = 60° + k.120°
  (de waarden −180° + k.360° zitten vervat in 60° + k.120°)
2^de manier :
cos 2x + cos x = 0   (SIMPSON toepassen)
2.cos \(\frac{3x}{2}\).cos \(\frac{x}{2}\) = 0
\(\cos\frac{3x}{2}=0\)   ∨   \(\cos\frac{x}{2}=0\)
\(\frac{3x}{2}\) = 90° + k.180°   ∨   \(\frac{x}{2}\) = 90° + k.180°
x= 60° + k.120°   ∨   x = 180° + k.360°
x= 60° + k.120°
  (de waarden 180° + k.360° zitten vervat in 60° + k.120°)
3^de manier :
cos 2x + cos x = 0  [ 1 + cos 2α = 2cos²α ]
2cos²x – 1 + cos x = 0
y = cos x  ∧  2y² + y − 1 = 0
y = cos x  ∧  (y + 1)(2y − 1) = 0
y = cos x  ∧  [ y = −1   ∨   y = 0,5]
cos x = −1   ∨   cos x = 0,5
x = 180° + k.360°   ∨   x = ±60° + k.360°
x = 60° + k.120°
Besluit : de oplossingen van   cos 2x + cos x = 0   worden (in radialen) gegeven door   \(\{\frac{\pi}{3}+k.\frac{2\pi}{3}\}\)   wat drie punten oplevert op de goniometrische cirkel