Trigonometry
     Trigonometric Equations
      Equations of the Type  a cos x + b sin x = c
         Introducing an auxiliary angle method
         Introducing an auxiliary angle method example
         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.
a)  Introducing an auxiliary angle method
Consider the constants a and b as rectangular coordinates of a point expressed by polar coordinates (r, j),
then,         a = r cos j   and    b = r sin j,
   
By substituting for a and b in the given equation 
a cos x + b sin x = c
obtained is,   r cos x cos j + r sin x sin j = or
using addition formula yields, since
obtained is the basic trigonometric equation whose solution is known.
Note that the given equation,  a cos x + b sin x = c  will have a solution if
 
it follows that the constants, a, b and c should satisfy relation  c2 < a2 + b2.
Introducing an auxiliary angle method example
Example:  Solve the equation,  sin x + 3 cos x = 1.
Solution:  Comparing corresponding parameters of the given equation with a cos x + b sin x = c it follows,
a = 3, b = 1 and  c = 1.
By substituting given quantities to the basic equation
or  x - 30 = + 60 + k 360  thus, solutions are, x = 90 + k 360 and  x = - 30 + k 360, k Z.
The same solution can be obtained using following procedure, from  sin x + 3 cos x = 1
             and   a cos x + b sin x = c | b
that means that we can introduce an auxiliary angle j that is
sin x sin 30 + cos x cos 30 = sin 30,     cos (x - 30) = 1/2
and this is the same basic equation obtained above.
b)  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 than 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.
Trigonometry contents B
Copyright 2004 - 2020, Nabla Ltd.  All rights reserved.