Самостоятельная работа для учащихся 10 класса по теме «Обратные тригонометрические функции»

Oбратные тригонометрические  функции.        Вариант  3.

1. Найдите значение выражения:                                                                                                                      а) arcsin;  б) arccos0;   в) arctg;    г) arcctg.

2 .Выразите значения данных функций через значения функции    y=arccosx:

a)arcsin б)arctg3;  в)arcctg.

3.Вычислите значения:                                                                                                                                                                                                                                                                        a) sin;  б)sin;     в) sin;                                                                                                                             г) arcsin;   д)2arctg+ arctg

4. Упростите выражение:    а) arccos   б)-arcsin;

5. Найдите значение выражения:      arcsin(sin(-6)).                                                     

6. Докажите справедливость раиенства:      tg 

7. Решите уравнение: 

                                 а) arcsinx=                

б) arcsinx = arcctgx;                                                                                                                                                         в) arccos(x+1) = arcctgx;                                                                                               г) arcsinx+arccosx =  arcsin.arcos;                                            д) arcsin(x +1) + arcos2x =0;                                                                                                                                             е) 4arccos x -4arccos x - 4arcsinx+=0;                                                               ж) arcsin(1+sinx)=arccos.

Oбратные тригонометрические  функции.             Вариант  1.

!. Найдите значение выражения:                                                                                                                      а) arcsin0;  б) arccos;   в) arctg1;    г)arcctg.

2 .Выразите значения данных функций через значения функции     у=arcsinx:                     а) arccos;  б) arctg ;   в) arcctg (-).             

3.Вычислите значения:                                                                                                                                                                                                                                                                        a) cos;  б)sin;     в) sin                                                                                                                             г) arcsin   д)2arctg2 + arcsin

4. Упростите выражение:    

                                 а)arcsinб)arcsin.

5. Найдите значение выражения:       arcsin(sin10).

6. Докажите справедливость раиенства:         cos(arcsin + arctg) =

7. Решите уравнение:  

                                 а)  arcsinx=-

б)  arcsinx = arccosx;                                                                                                                                                         в)  arcsin(x+1) = arctgx;                                                                                               г)  arcsin(1-2x) = 3 arcsin;                                                                             д) 2arcsinx + arcсos(1-x) =0;                                                                              е)  arcsin x + arcos x =                                                                  ж) cos(2arccosx) =arcsin(cosx).                    

Oбратные тригонометрические  функции.        Вариант  10.

1.Найдите значение выражения:                                                                                                                      а) arcsin(-);  б) arcos();   в) arctg(1); г)arcctg(-).

2 .Выразите значения данных функций через значения функции:

а)у=arcсtgx: arccos;  b) arctg ;   в) arcsin(-).             

3.Вычислите значения:                                                                                                                                                                                                                                                                        a) tg(3arcctg(-2));  б)ctg(arccos);     в) 2arctg4-arcsin;                                                                                                                      г)ctg(arccos); д)arccos +3arcsin.

4. Упростите выражение:

                                  а)arctg+arctg+arctg+arctg;б)arctg2-arctg.

5. Найдите значение выражения: arccos(cos8).

6. Докажите справедливость раиенства:

                                  cos(2arctg)-2sin(2arctg3)cos(2arctg)=0.

7. Решите уравнение: 

                                 а)arcctgx=;

б) arctg(x+1)+arctg(x-1)= ;                                                                                                                                                         в)arccos(4x-3) = 3arccosx;                                                                                               

г) 2arcsin2x = arccos7x;                                                                             д)arcctg(tg3x)=;                                                                              е)arcsin2x =arccos  ;                                                                ж)cos(2arccosx) =arcsin(cosx).                                          

Oбратные тригонометрические  функции.        Вариант  4.

!. Найдите значение выражения:                                                                                                                      а) arcsin;  б) arccos1;  в) arctg;    г)arcctg(-1).

2 .Выразите значения данных функций через значения функции:

а)у=arccosx: arcsin;  b) arctg(-2);   в) arcctg (-).             

3.Вычислите значения:                                                                                                                                                                                                                                                                        a) cos;  б)cos;     в) cos                                                                                                                             г) arcos(-0,9)+arcos(-)   д.)2arctg3 - arctg

4. Упростите выражение:   а)arccos       б)arcsin+ arctg(-2)+arcctg.

5. Найдите значение выражения:       arccos(cos12).

6. Докажите справедливость раиенства:     cos(2arcctg7) –sin(4arcctg3) =0.                   

7. Решите уравнение:  

                                 а)  arccosx=

б)  arcsinx = 2arctg;                                                                                                                                                  в)arccos(x-1) = arcctgx;                                                                                               г) arcsin(x-1)arccosx = -arcsin(x+1);                                                          д.)sin(3arccosx) =;                                                                            е)arctgx-arctgx -2arctgx+ = 0;                                                                ж)arccos(sinx+2sinx+2)=arcsin(1+cos2x).                  

Oбратные тригонометрические  функции.        Вариант  5.

!. Найдите значение выражения:                                                                                                                      а) arcsin1;  б) arccos;   в) arctg();    г)arcctg0.

2 .Выразите значения данных функций через значения функции:

а)у=arcsinx: arccos;  b) arctg ();   в) arcctg 2.             

3.Вычислите значения:                                                                                                                                                                                                                                                                        a) cos;  б)sin;     в) sin                                                                                                                             г) arcsin-arctg;   д)2arctg5 + arctg

4. Упростите выражение:            arctg+arctg;           3arcsin+arcsin.

5. Найдите значение выражения:        arcctg(ctg(-3).

6. Докажите справедливость раиенства: tg(2arccos - arcsin) = .

7. Решите уравнение:   а)arccosx=.

     б) arcsinx = 2arctg;                                                                                                                                                                  

     в)arctg(x-1)+ = 3arctg(x+1);                                                                                               

     г) arcsin(x-x-2) = 0;                                                                             д.)sin(2arccosx)  =;                                                                              е)arcsin + arcsin  =                                                                 ж)2arccos(2arctgx) =arccosx+arcsinx.  

Oбратные тригонометрические  функции.        Вариант  6.

!. Найдите значение выражения:                                                                                                                      а) arcsin();  б) arcos(-);   в) arctg();   г)arcctg(2-).

2 .Выразите значения данных функций через значения функции:

а)у=arcsinx: arccos;  b) arctg5;   в) arcctg (-).              

3.Вычислите значения:                                                                                                                                                                                                                                                                        a) tg;    б)cos;     в) sin                                                                                                                             г) arcsinarctg;         д)arccos + arcsin.

4. Упростите выражение: arctgarctg3;arccos- arcsin.

5. Найдите значение выражения: tg(2arccos).

6. Докажите справедливость раиенства: sin(2arctg) +tg( arcsin) =

7. Решите уравнение:   а)arccosx=-;

б)2arctg= arccosx;                                                                                                                                                         

в) arctg(x-2)+=3arctgx;                                                                                              

г) arccos =0;;                                                                             д)sin(arccosx)=;                                                                      е)arcsin + arcsin =;                                                                ж)sin(2arctgx)arctg(ctgx)=1.               

Oбратные тригонометрические  функции.        Вариант  7.

!. Найдите значение выражения:                                                                                                                      а) arcsin;  б) arccos;   в) arctg;    г)arcctg.

2 .Выразите значения данных функций через значения функции:

а)у=arctgx: arccos;  b) arctg 3;   в) arcctg (-).

3.Вычислите значения:                                                                                                                                                                                                                                                                        a) ctg(arcos(-));б)tg;     в) tg                                                                                                                             г) arctg+arctg;  д)arctg2 - arctg5.

4. Упростите выражение: arcsin arccos;2arctg2-arctg.

5. Найдите значение выражения: arctg(tg8).

6. Докажите справедливость раиенства: sin(4arcsin) - cos(2arcsin) =

 7. Решите уравнение:   а)arctgx=-;

б) 2arccos = arccosx;                                                                                                                                                         в)arctg(2x-3)=3arctgx;                                                                                               

г) arcsin(x-76+12)arcos(x-9x+21) = 0;                                                                             д)sin(arccosx)= + ;                                                                              е)arcsin  = arcos ;                                                               ж)cos(arcsinx)+ =arccos(sinx).         

исправлено

Oбратные тригонометрические  функции.        Вариант  8.

!. Найдите значение выражения:                                                                                                                      а) arcsin; б) arccos; в) arctg(-);г)arcctg.

2 .Выразите значения данных функций через значения функции:

а)у=arcsinx:     arccos;  б) arcsin ;   в) arcctg (-).             

3.Вычислите значения:                                                                                                                                                                                                                                                                        a) sin(arccos);  б)cos(3arcsin);     в)tg(2arccos);                                                                                                                              г) arctg + arctg;     д)2arctg2 + arcsin.

4. Упростите выражение:    a) arctg2-arcctg5;     б)4arctg-arctg.                                          

5. Найдите значение выражения: arcctg(ctg7).

6. Докажите справедливость раиенства: sin(4arcsin -cos(2arccos)= -

7. Решите уравнение:   а)arctgx=;

б) arctg(x+1)+arctg(x-1) = ;                                                                                                                                                        

в) arctg(2x-3)=3arctgx ;                                                                                             

г)сos(2arcsinx)=;                                                                            

д)arcsin(6x=13x-5) * arcos(5x-4x) =0

е)arcos(x+)=arcsin;                                                                                                                                 ж)sin(2arctgx)*tg(arcctgx) =1.               

Oбратные тригонометрические  функции.        Вариант  9.

1. Найдите значение выражения:                                                                                                                      а) arcsin(-);  б) arcos();   в) arctg(-1);    г)arcctg.

2 .Выразите значения данных функций через значения функции:

а)у=arcсtgx: arccos;  b) arctg ;   в) arcsin .     

4. Упростите выражение: arcsin+arccos;arcos(-)-arcctg(-).                             3.Вычислите значения:                                                                                                                                                                                                                               a)tg(arcsin);  б)cos(4arctg1);     в) ctg(2arcsin);                                                                                                                             г) 2arctg4+arcsin;д)arcsin + arcsin.

5. Найдите значение выражения: arcos(cos(-4)).

6. Докажите справедливость раиенства: cos(2arcsin -sin(4arccos) = -.

7. Решите уравнение:   а)arctgx=-;

б) arctg(x+1)+arctg(x-1) = -;                                                                                                                                                         в)arcsin(x+1) = arctgx;                                                                                               г) arcsin(x-2)+2 = 3 arcsinx ;                                                                            д)cos(arcsinx ) =;                                                                              е)arcsin  =arcos ;                                                                 ж)arccos =arcsin2x.                                               

Oбратные тригонометрические  функции.        Вариант  2.

!. Найдите значение выражения:                                                                                                                      а) arcsin;  б) arccos;   в) arctg0;    г)arcctg.

2 .Выразите значения данных функций через значения функции:                                 

а) у=arcsinx: arccos;  b) arctg ;   в) arcctg (-).             

3.Вычислите значения:                                                                                                                                                                                                                                                                        a) cos;  б)sin;     в) sin                                                                                                                             г) arcsin   д.)2arctg5 + arcsin

4. Упростите выражение:       a)arcsinarccos;        б)3arcsin+ arccos                         5. Найдите значение выражения:         arcsin(sin(-6)).

6. Докажите справедливость раиенства:         cos(2argtg2 –sin(4 arctg3) =

7. Решите уравнение: 

                                 а)  arcsinx=-;

б)  arcsinx = arctgx;                                                                                                                                                         в)  arcsin(x-1) = arctgx;                                                                                               г)  arcsin(3x+4) = 3 arccos;                                                                             д) 2arcsin(x=1) = arccos2x;                                                                              е)  4arccosx  - 4arccosx  - 6arcsin x + 3= 0.                                                                  ж)  cos(arcsinx) =arcos(sinx).                   

                  исправлен