Trigonometric Equations
      Basic Trigonometric Equations
         The equation  sin 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  sin x = a-1 < a < 1
To solve the equation we should find the arc x (or angle in radians) of which function value of sine equals a.
Infinite many arcs whose sine value equals a end in the points P and P¢that is,
x = arad + k 2p = arcsin a + k 2pk Z,  and
x = (p - arad) + k 2p = (p - arcsin a) + k 2pk Z,
there are an infinite number of solutions obtained by giving
different integer values to k. This is the set of the general 
solutions of the given trigonometric equation.
For k = 0 obtained are,
x0 = arcsin a  and   x0 = p - arcsin a,
  or   x0 = sin-1 a    and   x0 = p - sin-1 a,
the basic solutions.
The principal values of the inverse sine function, arcsin or sin-1, are those between - p/2 and p/2.
The above solutions of the equation  sin x = a-1 < a < 1 can concisely be written as 
  x = (-1)k arcsin a + k pk Z.  
For example,  a)  sin x = -1,                        b)  sin x = 0,                      c)  sin x = 1,
                               x = - p/2 + k 2p.               x = k p.                           x = p/2 + k 2pk Z.
Using a scientific calculator to obtain principal (or basic) value of the inverse sine function, arcsine, input value for a and press,  sin-1  (or INV sin  or  2nd sin).
Obtained is an arc (in radians) or angle (in degrees) between  - p/2 and p/2, depending on what measurement (DEG or RAD) was set before, by the DRG key.
Example:  Solve the equation,  
Solution:  Let first find the basic solutions of the equation. We should remember values of the trigonometric
functions of some special arcs (angles) like
Since the basic solutions for the equation sin x = a are,  x0 = arad  and   x0 = p - arad,  then
While from   x = x0 + k 2p  and   x = x0 + k 2pk Z,  follow
the general solutions of the given equation.
The basic solutions we see on the unit circle in the below figure while from the graph can be seen that the line y = - 3/2 intersects the sine function at infinite many points whose abscissas represent the general solutions.
Example:  Solve the equation,  sin x = 0.433266.
Solution:  Find the basic solution using a calculator. Press  DRG key to set DEG, meaning degree measurement of the corresponding angle, then input
          0.433266  sin-1  (or INV sin)    =>     x0 = a = 25.67500799 = 2540′30″,    
                                                                x0 = 180 - a = 154.324992 = 15419′30″.
To transform the decimal degrees to deg., min. and sec. form, press  DMS.
If the basic solution should be presented in radians, set RAD measurement.
It is also possible an angle expressed in degrees convert straight to radians. While display shows decimal degrees press DRG  so obtained are,
x0 = arad = 0.448113424rad    and    x0 = p - arad = 2.693479229rad.
Thus, general solution presented by angles,     x = x0 + k 360 = 2540′30″ + k 360,  k Z,
                                                        and      x = x0 + k 360 = 15419′30″ + k 360,  k Z,
                                               or by arcs,     x = x0 + k 2p = 0.448113424rad + k 2p,  k Z,
                                                        and     x = x0 + k 2p = 2.693479229rad + k 2p,  k Z.
Trigonometry contents B
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