Solving Trigonometric Equations by Jeqtax

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									Lesson 5.3


        Solving
 Trigonometric Equations
             Solving
      Trigonometric Equations
To solve trigonometric equations:

 Use standard algebraic techniques learned in Algebra II.

 Look for factoring and collecting like terms.
 Isolate the trig function in the equation.

 Use the inverse trig functions to assist in determining
 solutions.
                       Solving
                Trigonometric Equations

                   For all problems,
                  The solution interval
                        Will be
                         [0, 2)

You are responsible for checking your solutions back into the original problem!
                Solving
         Trigonometric Equations
Solve:     2cos x 1  0
Step 1: Isosolate cos x using algebraic skills.

          2cos x  1
           cos x  1
                          2
Step 2: Determine in which quadrants cosine is positive. Use the inverse
function to assist by finding the angle in Quad I first. Then use that angle
as the reference angle for the other quadrant(s).
                 QI       QIV               Note: cosine is positive in
                                            Quad I and Quad IV.
                  5                      Note: The reference angle is /3.
           x         ,
                 3        3
                 Solving
          Trigonometric Equations
Solve:    tan x  1  0
                 2

Step 1:   tan x  1
                 2

          tan x   1
             2
                                                     Note: Since there is a  , all four quadrants

           tan x  1                                hold a solution with /4 being the reference
                                                     angle.

  Step 2:                                      QIV
                 Q1       QII       QIII
                  3 5 7
            x        ,         ,          ,
                 4 4                4          4
                   Solving
            Trigonometric Equations
    Solve: cot x cos x  2cot x
                    2



Step 1:     cot x cos2 x  2cot x  0
            cot x  cos2 x  2  0
            cot x  0 or cos 2 x  2  0
                              cos2 x  2
                            cos2 x   2
                             cos x   2


                 3         x           Note: There is no solution here because 2
Step 2:    x       ,                      lies outside the range for cosine.
                2       2
            Solving
     Trigonometric Equations
      Try these:       Solution
                           3 7
1.   tan x  1  0    x     ,
                            4 4
                            2 4 5
2.   sec x  4  0
        2             x
                           3
                               ,
                                   3
                                           ,
                                               3
                                                   ,
                                                       3
                                    5                7 11
3.   3tan x  tan x
            3
                      x  0,
                                   6
                                       ,
                                           6
                                               , ,
                                                        6
                                                          ,
                                                            6
                Solving
         Trigonometric Equations
Solve:   2sin 2 x  sin x  1  0
          2sin x 1sin x 1  0         Factor the quadratic equation.

         2sin x  1  0 or sin x  1  0    Set each factor equal to zero.

                   1
         sin x                sin x  1   Solve for sin x
                   2
              7 11                  
         x      ,              x          Determine the correct quadrants
               6     6                2     for the solution(s).
                Solving
         Trigonometric Equations
Solve:   2sin 2 x  3cos x  3  0

         2 1  cos2 x   3cos x  3  0   Replace sin2x with 1-cos2x
         2  2cos2 x  3cos x  3  0       Distribute
         2cos2 x  3cos x 1  0           Combine like terms.
         2cos2 x  3cos x  1  0           Multiply through by – 1.
          2cos x 1 cos x 1  0        Factor.
         2cos x  1  0 or cos x  1  0    Set each factor equal to zero.
                   1
         cos x                cos x  1    Solve for cos x.
                   2
                5
         x            ,       x0          Determine the solution(s).
               3           3
                Solving
         Trigonometric Equations
Solve:   cos x  1  sin x                    Square both sides of the equation
          cos x  1         sin x 
                        2                 2
                                              in order to change sine into terms
                                              of cosine giving only one trig
                                              function to work with.

         cos2 x  2cos x  1  sin 2 x        FOIL or Double Distribute
         cos2 x  2cos x  1  1  cos2 x     Replace sin2x with 1 – cos2x
         2cos2 x  2cos x  0                 Set equation equal to zero since it is a
                                              quadratic equation.
         2cos x  cos x 1  0               Factor
         2cos x  0 or cos x  1  0          Set each factor equal to zero.
         cos x  0     cos x  1             Solve for cos x
                 3
          x     ,X                  x      Determine the solution(s).
                2 2
  Why is 3/2 removed as a solution?               It is removed because it does not
                                                   check in the original equation.
                      Solving
               Trigonometric Equations
                                            1
      Solve:    cos 3x 
                                            2
               No algebraic work needs to be done because cosine is already by itself.
Solution:      Remember, 3x refers to an angle and one cannot divide by 3 because it
               is cos 3x which equals ½.
               Since 3x refers to an angle, find the angles whose cosine value is ½.
                       5
               3x          ,                       Now divide by 3 because it is angle equaling angle.
                        3           3
                     5                            Notice the solutions do not exceed 2. Therefore,
               x       ,
                    9           9                   more solutions may exist.
                       5 7 11                   Return to the step where you have 3x equaling
               3x          ,           ,       ,   the two angles and find coterminal angles for
                     3 33  3
                                                    those two.
                   5 7 11
                x ,  ,  ,                          Divide those two new angles by 3.
                  9 9 9    9
                       Solving
                Trigonometric Equations
     5 7 11 13 17                            The solutions still do not exceed 2.
3x  ,  ,  ,   ,   ,                               Return to 3x and find two more
    3 3 3    3   3   3                             coterminal angles.
       5 7 11 13 17
 x    ,  ,  ,   ,   ,                         Divide those two new angles by 3.
      9 9 9    9   9   9
     5 7 11 13 17 19 23 The solutions still do not exceed 2.
3x  ,  ,  ,   ,   ,    ,   ,   Return to 3x and find two more
    3 3 3    3   3   3    3   3 coterminal angles.

       5 7 11 13 17 19
x     ,  ,  ,   ,   ,   ,                     Divide those two new angles by 3.
      9 9 9    9   9   9   9
 Notice that 19/9 now exceeds 2 and
 is not part of the solution.
                                                5 7 11 13 17
Therefore the solution to cos 3x = ½ is   x       ,       ,       ,       ,         ,
                                               9       9       9       9        9          9
              Solving
       Trigonometric Equations
     Try these:              Solution
1.   4sin 2 x  2cos x  1   x  5.4218
                                 
2.   csc x  cot x  1       x
                                 2
                   3             2 5 5 11
3.    sin 2 x              x       ,       ,       ,
                  2               3       6       3       6
        x   2                     
4.   cos                    x
        2  2                      2
            Solving
     Trigonometric Equations
What you should know:


1. How to use algebraic techniques to solve
   trigonometric equations.

2. How to solve quadratic trigonometric equations
   by factoring or the quadratic formula.

3. How to solve trigonometric equations involving
   multiple angles.

								
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