Equações trigonometricas

Trigonometric equations are equalities that involve trigonometric functions of unknown arcs. The resolution of these equations consists of a single process that uses techniques of reduction to simpler equations. We will address the concepts and definitions of the equations in the form = cosx.

Trigonometric equations in the form α = cosx has solutions in the interval -1 ≤ x ≤ 1 The determination of the values of x that satisfy this equation type obey the following property:. If two arcs have equal cosines, then they are côngruos or replementares.

Consider x = α a solution of the equation cos x = α. Other possible solutions are to côngruos α bow or archery bows - α (or arc 2π - α). So: cos x = cos α. Note the trigonometric representation in the cycle:

We conclude that: 

x = α + 2kπ with k Є Z or x = - α + 2kπ with k Є Z 

example 1 

Solve the equation cos x = √ 2/2. 

By the trigonometric ratios table we have √ 2/2 corresponds to an angle of 45 º. then: 

cos x = √ 2/2 → cos x = π / 4 (π / 4 = 180/4 = 45) 

Thus, the equation cosx = √ 2/2 has as a solution to all côngruos arc π / 4 or-π / 4 or arcs 2π - π / 4 = 7π / 4. Note illustration:
We conclude that the possible solutions of the equation cos x = √ 2/2 are:
x = π / 4 + 2kπ with k Є Z or x = - π / 4 + 2kπ with k Є Z