4 The solutions of Cos x 1 in the interval 0 lessthanorequal

4

The solutions of Cos x= -1, in the interval 0 lessthanorequalto x lessthanorequalto 2pi 5.the solutions of cos 4x=-1, in the interval 0 lessthanorequalto x lessthanorequalto 2pi

Solution

y = cosx for 0 <=x< 2pi

1) cosx = 1/2

x = cos^-1(1/2)

x= pi/3 , 2pi -pi/3 = pi/3 , 5pi/3

2) cos3x = 1/2

3x = cos^-1(1/2)

3x = pi/3 +2kpi ---> x= pi/9 +2kpi/3

3x = 5pi/3 +2kpi ---> x= 5pi/9 +2kpi/3

k=0 ; x= pi/9 , 5pi/9

k=1 ; x= pi/9 +2pi/3 = 7pi/9 ; x= 5pi/9 +2pi/3 = 11pi/9

k=2 ; x= pi/9 +4pi/3 = 13pi/9 ; x= 5pi/9 +4pi/3 =17pi/9

k=3 ; x= pi/9 +pi = 10pi/9 ; x = 5pi/9 +pi =14pi/9

In the interval 0<= x<2pi .Solution : ( x=  pi/9 , 5pi/9 , 7pi/9 ,11pi/9 ,13pi/9 , 17pi/9, 10pi/9 , 14pi/9)

3) All solutions is the general solution.

x= pi/9 +2kpi/3

x= 5pi/9 +2kpi/3

4) cosx = -1 in the interval 0<=x <2pi

x= cos^-1(-1) = pi

solution in the given interval : x= pi