Trigonometry
     Trigonometric Equations
      Equations of the Type   a cos x + b sin x = c
         Introducing new unknown  t = tan x/2
         Introducing new unknown  t = tan x/2 example
Equations of the type  a cos x + b sin x = c
To solve the trigonometric equations which are linear in sin x and cos x, and where, a, b, and c are real 
numbers we can use the two methods,
a)  introducing an auxiliary angle, and  b)  introducing new unknown.
Introducing  new unknown  t = tan x/2
If in the equation a cos x + b sin x = c we substitute the sine and cosine functions by tan x/2 = t  that is,
the equation becomes  and after rearranging (a + c) t2 - 2b t + (c - a) = 0.
Obtained quadratic equation will have real solutions t1,2 if its discriminant is greater then or equal to zero,  
that is if   (-2b)2 - 4 (a + c)(c - a) > 0   or   c2 < a2 + b2,   which is earlier mentioned condition.
If this condition is satisfied, the solutions, t1 and t2 can be substituted into tan x/2 = t1 and  tan x/2 = t2.
Thus, obtained are the basic trigonometric equations.
Introducing new unknown  t = tan x/2 example
Example:  Solve the equation,  5 sin x - 4 cos x = 3.
Solution:  Given equation is of the form a cos x + b sin x = c therefore parameters are, a = - 4, b = 5 and  
c = 3, after introducing new unknown tan x/2 = t and substituting the values of the parameters into equation
(a + c) t2 - 2b t + (c - a) = 0    gives   (- 4 + 3) t2 - 2 5 t + [3 - (- 4)] = 0
 or      t2 + 10t - 7 = 0,    t1,2  = - 5 + 25 + 7 = - 5 + 42.
Obtained values for variable t we plug into substitutions,
         tan x/2 = t1,   x/2 = tan-1 (t1 or  x = 2arctan(- 5 - 42)
                                                             x = 2 (- 843821 + k 180) = - 1691642 + k 360,
         tan x/2 = t2,   x/2 = tan-1 (t2 or  x = 2arctan(- 5 + 42)
                                                             x = 2 (331756 + k 180) = 663553 + k 360.
The same result we obtain using the method of introducing the auxiliary angle j. Plug the given parameters
a = - 4, b = 5 and c = 3 into  tan j = b/a,  tan j = 5/(- 4)  =>  j = -512024,   cos j = 0.624695,  
then from the equation  cos (x + 512024″) = (3/- 4) 0.624695
       or  x + 512024 = arccos [(3/- 4) 0.624695],
thus,  x + 512024 = + 1175618 + k 360   =>   x = + 1175618 - 512024 + k 360.
Trigonometric equations of the form a cos x + b sin x = c we do not solve using the identity
since that way given equation becomes quadratic with four solutions but only two of them satisfy it.
We will solve the equation from the previous example using this method anyway.
Example:  Solve the equation,  5 sin x - 4 cos x = 3 by substituting 
Solution:  Squaring both sides of an equation may introduce extraneous or redundant (not needed) solutions.
By plugging the results into given equation show that only solutions b) and c) satisfy the equation what match with previous results obtained using another two methods.
Pre-calculus contents G
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