Trigonometry
Trigonometric Equations
sin (bx + c) = m,  -1 <  m < 1,      cos (bx + c) = m,  -1 <  m < 1,
cos (bx + c) = m,  -1 <  m < 1, example
The Equations,  sin (bx + c) = m,  -1 <  m < 1,       cos (bx + c) = m,  -1 <  m < 1,
The given equations can be written as F (bx + c) = m  where F substitutes a trigonometric function, x is an arc to be calculated and m is a value of a given trigonometric function.
To every trigonometric function we can determine an arc, a + k · P of which function value equals m that is   F (a + k · P) = m, where a = x0  is the basic solution, and P is the period, then
F(bx + c) = F(a + k · P)    or   bx + c = a + k · P,  thus
Thus, from obtained general solution we can write a common solutions for every given equation,
The Equation  cos (bx + c) = m,  -1 <  m < 1, example
Example:  Find the solutions of the equation,  2cos (4x - 30°) + Ö3 = 0.
Solution:  Rewrite the equation to the form cos (bx + c) = m, that is  cos (4 x - 30°) = - Ö3/2
it follows that    cos (4 x - 30°) =  cos (± 150° + k · 360°)
and             4 x - 30°± 150° + k · 360°
so,       x = 45° + k · 90°   and    x = - 30° + k · 90°,  kÎ Z.
The same results we get by substituting the values, b = 4, c = - 30° and  m = - Ö3/2, into
x0 = a = cos-1 m = cos-1 (- Ö3/2)  = 150°  and  x0 - a = -150°
then, using the common solution formulas obtained are
Pre-calculus contents G