Treat a single trigonometric ratio as an algebraic quantity. If AB=0, either A=0 or B=0. Dividing by a factor that may be zero removes that entire branch. Factorisation records both branches before any division. 1. Move all terms to one side. 2. Factorise in one ratio. 3. Solve every factor. 4. Reject impossible ratio values. 5. List distinct angles in the original interval. Example: sin2x+sin x=0 becomes sin x(sin x+1)=0. For 0 <= x<360, the roots are 0,180,270. Exam check: dividing this example by sin x would lose 0 and 180. A repeated algebraic factor does not give a repeated angle in the answer set.

Treat a single trigonometric ratio as an algebraic quantity. If \(AB=0\), either \(A=0\) or \(B=0\).

Dividing by a factor that may be zero removes that entire branch. Factorisation records both branches before any division.

1. Move all terms to one side. 2. Factorise in one ratio. 3. Solve every factor. 4. Reject impossible ratio values. 5. List distinct angles in the original interval.

Example: \(\sin^2x+\sin x=0\) becomes \(\sin x(\sin x+1)=0\). For \(0^\circ\le x<360^\circ\), the roots are \(0^\circ,180^\circ,270^\circ\).

Exam check: dividing this example by \(\sin x\) would lose \(0^\circ\) and \(180^\circ\). A repeated algebraic factor does not give a repeated angle in the answer set.

Factorisation without losing roots

Factorisation without losing roots