Equations reducible to homogeneous ones

Not all equations with quadratic terms in $\sin$ and $\cos$ are already homogeneous: sometimes a constant term appears. A simple trick based on the fundamental identity lets us reduce them to homogeneous form.

Observation — Equations reducible to homogeneous ones

If an equation also contains a “constant” term — of the type $A\sin^2 x + B\sin x\cos x + C\cos^2 x + D = 0$ — it can be brought to homogeneous form by multiplying the term $D$ by $\sin^2 x + \cos^2 x = 1$, which turns it into $D\sin^2 x + D\cos^2 x$ and distributes it over both quadratic terms.

Links

Topics: Trigonometric equations
Concepts: Homogeneous equation
Functions: Cosine · Sine
Skills: Solving equations · Using formulae