Definition and solution method

Second-degree homogeneous equations in $\sin$ and $\cos$ are recognised because every monomial has the same degree. The solution method turns them into a second-degree equation in $\tan x$.

Definition — Second-degree homogeneous equation

A trigonometric equation is said to be homogeneous of second degree in $\sin$ and $\cos$ if it is of the type $A\sin^2 x + B\sin x\cos x + C\cos^2 x = 0.$ All the monomials have the same total degree in $\sin$ and $\cos$: two.

In summary — Solution method

1. Check whether $\cos x = 0$ is a solution, by substituting $x=\pi/2+k\pi$ directly.
2. Divide the equation by $\cos^2 x$ (possible because the case $\cos x = 0$ has already been handled): $A\tan^2 x + B\tan x + C = 0.$
3. Set $t = \tan x$ and solve a second-degree equation in $t$.
4. Go back to $x$ with $x = \arctan t + k\pi$.

Example

Solve $\sin^2 x - 3\sin x\cos x + 2\cos^2 x = 0$.

Case $\cos x = 0$: substituting $x=\pi/2$ gives $1 - 0 + 0 = 1\ne 0$. Not a solution.

Divide by $\cos^2 x$: $\tan^2 x - 3\tan x + 2 = 0.$ With $t=\tan x$: $\ t^2 - 3t + 2 = 0 \iff (t-1)(t-2) = 0 \iff t=1 \ \vee\ t=2$.

Back to $x$: $\ x = \arctan 1 + k\pi = \dfrac{\pi}{4} + k\pi$, or $x = \arctan 2 + k\pi$.

Links

Topics: Trigonometric equations
Concepts: Trigonometric equation · Homogeneous equation
Functions: Cosine · Sine · Tangent
Methods: Trigonometric equations
Skills: Reasoning by cases · Solving equations · Using formulae