Trigonometric equations are equalities that evolve one or more trigonometric functions of unknown arcs. To solve trigonometric equations there is no single process, what we should do is try to reduce them to simpler equations, like senx = α,
cosx = α and tgx = α, called fundamental equations. From the three equations mentioned, we will address the concepts and ways of solving the equation senx = α.
Trigonometric equations in form senx = α have solutions in the range –1 ≤ x ≤ 1. Determining the values of x that satisfy this type of equation will obey the following property: If two arcs have equal sines, then they are congruent or supplementary.
let's consider x = α a solution of the equation sin x = α. The other possible solutions are the arcs congruent to the arc α or to the arc π – α. Then: sin x = sin α. Note the representation in the trigonometric cycle:
We concluded that:
x = α + 2kπ, with k Є Z or x = π – α + 2kπ, with k Є Z
Example
Solve the equation: sin x = √3/2
We know from the table of trigonometric ratios that √3/2 corresponds to the sine of the 60° angle. Then:
sin x = √3/2 → sin x = π/3 (π/3 = 180º/3 = 60º)
Thus, the equation senx = √3/2 has as a solution all the arcs congruent to the arc π/3 or to the arc π – π/3. Note the illustration:
We conclude that the possible solutions of the equation sin x = √3/2 are:
x = π/3 + 2kπ, with k Є Z or x = 2π/3 + 2kπ, with k Є Z
by Mark Noah
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/equacoes-tipo-sen-x-a.htm