Тема урока:  Решение тригонометрических уравнений

                                                t = ± arccos a + 2Пn, n ? Z

                                               Частные случаи:

cos x = 0                        cos x = - 1                           cos x = 1

x = П/2 +П n, n ? Z      x = П + 2 Пn, n ? Z       x = 2Пn, n ? Z

                                                  2. sin t = a, aЄ[-1;1]

                                              t = (-1)ⁿ arcsin a + Пn, n ? Z

                                              Частные случаи:

sin x = 0                           sin x = -1                            sin x = 1

х = П n, n ? Z                х = - П/2 + 2Пn, n ? Z      х = П/2 + 2Пn, n ? Z

         3. tg t = a                                    4. ctg t = a

          t = arctg a + Пn, n ? Z                      t = arcctg a +  Пn, n ? Z

Примеры. Решите уравнение:

1)  cos x = - ½                              2)  sin 2x = - 1

              х = ± arcos + 2Пn                          2x = - + 2Пn │x

              х = ± + 2Пn, n ? Z                               x = - + Пn, n ? Z     

          Ответ: ± + 2Пn, n ? Z                           Ответ: - + Пn, n ? Z                                       

                 3) 2 sin² x + sinx  - 1 = 0

                       sinx = y, y Є[-1;1]

                       2у² + у - 1 = 0

             D = b² - 4ac = 1 + 8 = 9

              y =  =  = [

                       sinx = -1                и        sinx =  

                       x = -  + 2πn, nЄZ                  x = (-1)ⁿ  + πn, nЄZ

                      Ответ: -  + 2πn; (-1)ⁿ  + πn, nЄZ

                 4) 6 sin² x + 5cosx  - 2 = 0                  sin² x + cos²x =1 =>  

                      6 (1 - cos²x ) + 5cosx  - 2 = 0           => sin² x = 1 - cos²x 

                      6 - 6cos²x  + 5cosx  - 2 = 0 

                      6cos²x  - 5cosx  - 4 = 0 

                       cosx  = y, y Є[-1;1]

                       6у² -  5у - 4 = 0

             D = b² - 4ac = 25 + 96 = 121 = 11²

              y =  =  = [

                       cosx = -                     

                       x = ±  + 2πn, nЄZ                 

                      Ответ: ±  + 2πn, nЄZ                 

                 5) 3 sin² x - 4 sinx cosx + cos²х = 0│: cos² x        cos x ≠ 0

                       3 tg²x – 4 tg x + 1 = 0                                   x ≠ π/2 + πn, n Є Z

                       tg x = у                                                            tgα =

                       3у² - 4 у + 1 = 0

                       D = 4, у = ⅓, у = 1

                       tg x = 1                и        tg x = ⅓ 

                       x = π/4 + πn, nЄZ               x = arctg ⅓ + πn, nЄZ

                      Ответ: π/4 + πn; arctg ⅓ + πn, nЄZ

                  6) 6 sin² x + 4 sinx cosx = 1                                      sin²x + cos²x = 1

                      6 sin²x + 4 sinx cosx = sin²x + cos²x                      

                      5 sin²x + 4 sinx cosx – cos²x = 0                     cos x ≠ 0

                      5 tg²x + 4 tg x - 1 = 0                                    x ≠ π/2 + πn, n Є Z

                       tg x = у

                       5у² + 4 у - 1 = 0

                       D = 36, у = - 1, у = 1/5

                       tg x = - 1                      и             tg x = 1/5

                      x = - π/4 + πn, n Є Z                       x = arctg 1/5 + πn, n Є Z

                      Ответ:  - π/4 + πn;   arctg 1/5 + πn, n Є Z