Trigonometric Equations
      Basic Trigonometric Equations
         The equation  cos x = a
Trigonometric equations
An equation that involves one or more trigonometric functions, of an unknown arc, angle or number, is called trigonometric equation.
Basic trigonometric equations
The equation  cosx = a-1 < a < 1
The solutions of the equation are arcs x whose function's value of cosine equals a.
Infinite many arcs whose cosine value equals a end in points, P
and P, that are
  x = arad + k 2p  and  x = - arad + k 2pk Z.  
This is the set of the general solutions of the given equation. 
For k = 0 follows the basic solutions of the equation
  x0 = arad   and  x0 = - arad.  
Therefore, if  cos x = a-1 < a < 1 then,  x = + arad + k 2p = + arccos ak Z.
For example if,   a = -1, then,     cos x = -1      =>       x = p + k 2pk Z,
                        a = 0                 cos x = 0        =>       x = p/2 + k pk Z,
                  or   a = 1                 cos x = 1        =>       x = k 2pk Z.
Since cosine function passes through all values from range -1 to 1 while arc x increases from 0 to p, one of the arcs from this interval must satisfy the equation cos x = a.
This arc, denoted x0, we call the basic solution.
Thus, the basic solution of the equation cos x = a-1 < a < 1 is the value of inverse cosine function,         x0 = arccos a   or   x0 = cos-1 a,  that is, an arc or angle (whose cosine equals a) between 0 and p which is called the principal value.
Scientific calculators are equipped with the arccos (or cos-1) function which, for a given argument between     -1 and 1, outputs arc (in radians) or angle (in degrees) from the range x0 [0, p].
Example:  Solve the equation,  cos x = - 0.5.
Solution:  In the unit circle in the below figure shown are the two arcs, of which cosine value equals - 0.5, that represent the basic solutions of the given equation
x0 = 120     or    x0 = -120
while the abscissas of the intersection points of the line y = - 0.5 with the graph of cosine function represent the set of the general solution
x = + 120 + k 360     or    x = + 2p/3 + k 2pk Z.
The same results we obtain by using calculator if we set DEG then input
- 0.5  INV  cos  (or cos-1)     =>     x0 = 120     and    x0 = -120  that are the basic solutions.
Or we input the same while calculator is set in RAD mode to get the arc in radians that is
x0 = 2.094395102rad = 2p/3rad.
Trigonometry contents B
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