Systems of trigonometric equations and inequalities

A system of trigonometric equations and inequalities involves finding all values of the variable (usually $x$) that satisfy several equations and/or inequalities involving trigonometric functions (like $\sin x$, $\cos x$, $\tan x$) at the same time. For example, solving $\sin x = \frac12$ and $\cos x > 0$ together means finding all $x$ that make both statements true.

Important properties

• Trigonometric functions are periodic, so solutions often repeat in regular intervals.

• You may need to use trigonometric identities to simplify or combine equations.

• The solution to a system is the intersection of the solution sets for each equation or inequality.

• Inequalities with trigonometric functions often require analyzing the sign or range of the function over intervals.