Solving Trigonometric Equations

1. 2 sin²x = 3 sin x - 1
1. Set equation equal to 0:  2 sin²x -3 sin x + 1 = 0
2. Factor:                      (2 sin x - 1)(sin x - 1) = 0
3. Set each factor equal to 0:  2 sin x - 1 = 0              sin x - 1 = 0

                                               2 sin x = 1                   sin x = 1
                                                  sin x = 1/2    arcsin(sin x) = arcsin 1
                                      arcsin(sin x) = arcsin(1/2)          x = 90^o
                                                                           x = 30^o, 150^o
2. 3 cos²x + 8 cos x + 1 = 0
1. Since this quadratic equation does not factor, use the quadratic formula:

    cos x =   -b + \/(b² - 4ac)    or     cos x =   -b - \/(b² - 4ac)
                                   2a                                        2a
2. cos x = -8 + \/(64 - 4(3)(1))     or     cos x =  -8 + \/(64 - 4(3)(1))

                          2(3)                                                   2(3)
cos x = -8 + \/52                              cos x = -8 - \/52
                 6                                                      6
cos x =  -.1315                                 cos x = -2.535

      x = 97.56^o, 262.44^o                         x = empty set (-1 £ cos x £ 1)

2 sin²x - cos x = 1

1. In order to have an equation in one trigonometric function, use an identity to

change one of them.  In this case, use sin²x = 1 - cos²x.
so  2(1 - cos²x) - cos x = 1
2. Use algebra to finish:
1. Distributive Property:              2 - 2 cos²x - cos x = 1
2. Set equation equal to 0:           1 - 2 cos²x - cos x = 0
3. Put in descending order:        -2 cos²x - cos x +  1= 0
4. Multiply by -1:                          cos²x + cos x - 1 = 0
5. Factor:                            (2 cos x - 1)(cos x + 1) = 0
6. Set each factor equal to 0 and solve for x:

                      2 cos x - 1 = 0           cos x = -1
                            2 cos x = 1                 x = 180^o
                                        cos x = 1/2
                                    x = 60^o, 300^o

Problems