TMUA Trigonometric Equations

TMUA trigonometric equations almost always pair a trig substitution with a counting question: “how many solutions does this equation have in this interval?” The bookkeeping — translating an interval in $\theta$ to an interval in $3\theta$ or $2\theta + \tfrac{\pi}{4}$ and counting cosine / sine solutions there — is the actual test.

• Multiple-angle substitution. Set $\phi = 3\theta$ (or whatever the argument is). Convert the interval in $\theta$ to the corresponding interval in $\phi$. Count solutions of the simpler equation in $\phi$, then convert each back.
• Range of cosine / sine. $-1 \le \cos x \le 1$. An equation $\cos x = k$ with $|k| > 1$ has no solutions; $|k| = 1$ has solutions only at the unique angle.
• Identities. $\cos^2 + \sin^2 = 1$; $\sin(2x) = 2\sin x \cos x$; $\cos(2x) = 1 - 2\sin^2 x = 2\cos^2 x - 1$ when needed.
• Reading the interval. “In $0^\circ \le \theta \le 180^\circ$” is inclusive of both endpoints. Test endpoint values explicitly.

The move. Substitute first, count second. Once the argument is a clean single variable, the interval extends accordingly and the count comes from reading the cosine / sine graph.