The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.
Trigonometric equations that become quadratics
Use one trigonometric ratio as an algebraic variable.
Reject impossible sine and cosine values and retain both signs when appropriate.
Count distinct angle solutions, including endpoint and repeated-root cases.
Solve the algebraic equation in \(u=\sin x\), \(\cos x\) or \(\tan x\). Sine and cosine must lie in \([-1,1]\); tangent is not subject to that bound. Each admissible ratio still needs every angle in the stated interval. Repeated algebraic roots do not create additional distinct angles.