A trigonometric equation is an equation containing one or more trigonometric functions of the unknowns. When inverse trigonometric functions are used to solve it, the complete solution set must account for the relevant branches and periods.
For a rational function 
with constant coefficients,
an equation 
can be transformed by the Weierstrass
substitution 
into
 |
The half-angle formulas give this reduction (Feldman et al. 2024, pp. 125-126). Multiple-angle formulas put finite rational expressions in the sine, cosine, and tangent of integer multiples of a common angle into this form.
For real 
, each finite real solution
gives 
,
where 
and 
is the principal value of the inverse
tangent. The omitted angles 
must be checked in the original equation.
Clearing denominators gives a polynomial
equation, while retaining all domain restrictions
of the original expression.
