Trigonometric equations are equalities that involve trigonometric functions of unknown arcs. Solving these equations is a unique process that uses techniques of reduction to simpler equations. Let's cover the concepts and definitions of equations in the form cosx = a.
Trigonometric equations in the form cosx = α have solutions in the interval –1 ≤ x ≤ 1. Determining the values of x that satisfy this type of equation will obey the following property: If two arcs have equal cosines, then they are congruent or complementary..
Let x = α be a solution of the equation cos x = α. The other possible solutions are the arcs congruent to the arc α or to the arc – α (or to the arc 2π – α). So: cos x = cos α. Note the representation in the trigonometric cycle:
We concluded 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.
From the table of trigonometric ratios, que2/2 corresponds to an angle of 45º. Then:
cos x = √2/2 → cos x = π/4 (π/4 = 180º/4 = 45º)
Thus, the equation cosx = √2/2 has as a solution all the arcs congruent to the arc π/4 or –π/4 or even 2π – π/4 = 7π/4. Note the 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
Example 2
Solve the equation: cos 3x = cos x
When the 3x and x arcs are congruent:
3x = x + 2kπ
3x - x = 2kπ
2x = 2kπ
x = kπ
When the 3x and x arcs are complementary:
3x = –x + 2kπ
3x + x = 2kπ
4x = 2kπ
x = 2kπ/4
x = kπ/2
The solution of the equation cos 3x = cos x is {x Є R / x = kπ or x = kπ/2, with k Є Z}.
by Mark Noah
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/equacoes-tipo-cos-x-a.htm
