The sign-study method

If the inequality is not elementary, it is factorised using the formulae of the previous chapter (addition, double-angle, sum-to-product) and the sign of the individual factors is studied. The strategy is the same as for the fractional inequalities or those with sign study seen in Year 2: reduce everything to a product (or quotient) of elementary factors whose sign can be read.

In summary — General procedure

1. Factorise using identities, double-angle formulae, sum-to-product formulae or algebraic substitutions.
2. Study the sign of each factor on the unit circle, marking the corresponding arcs with "$+$" and "$-$".
3. Build a single sign table with all the factors.
4. Read the solution from the table, identifying the arcs where the overall sign satisfies the inequality.

The key step is the first: reducing the expression to a product of elementary factors. Each factor is then an elementary inequality, whose sign is read on the circle exactly as seen for $\sin$, $\cos$ and $\tan$.

Links

Topics: Trigonometric inequalities
Concepts: Trigonometric inequality · Double-angle formulae · Factorisation · Sign study · Sign table
Functions: Cosine · Sine
Skills: Reasoning by cases · Solving inequalities · Using formulae