Trigonometry
     Trigonometric Equations
      The Equations,    sin (bx + c) = m,  -1 <  m < 1,      cos (bx + c) = m,  -1 <  m < 1,
         The Equation  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) + 3 = 0.
Solution:  Rewrite the equation to the form cos (bx + c) = m, that is  cos (4 x - 30) = - 3/2
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 90k Z.
The same results we get by substituting the values, b = 4, c = - 30 and  m = - 3/2, into
x0 = a = cos-1 m = cos-1 (- 3/2)  = 150  and  x0 - a = -150 
then, using the common solution formulas obtained are
Pre-calculus contents G
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