Complex trigonometric equations require systematic solution strategies combining algebraic and trigonometric techniques.
Solution Methods
- Linear Form: asinx + bcosx = c
Convert to Rsin(x + α) where R = √(a² + b²)
- Quadratic Form: Use substitution (u = sinx)
- Multiple Angles: Apply double-angle identities
Equation Solution
Solve: 2sin²x - 3sinx + 1 = 0
Let u = sinx → 2u² - 3u + 1 = 0
Solutions: u=1 ⇒ x=π/2 + 2πn
u=½ ⇒ x=π/6 + 2πn or 5π/6 + 2πn
Special Cases
- Homogeneous Equations: asinx + bcosx = 0 ⇒ tanx = -b/a
- Symmetrical Forms: sinx + cosx = k ⇒ use sin(x + π/4)