Your starting equation is 

    7cos(4x) = 5.       (1)


As you understand, it is EQUIVALENT to

    cos(4x) = .       (2)


From (2), you have these two series of solutions

    4x =  +  = 0.775193 +                         (3)

and 

    4x =  +  =  + ,     (4)


where n = 0, +/-1, +/-2, +/-3, +/-4, . . . is any integer number.



From (3), dividing both its sides by 4, you have

    x = 0.193798 +  = 0.193798 + .

    First four positive values for n= 0, 1, 2 and 3 are (rounded to 4 decimals)  x = 0.1938,  1.7646,  3.3354,  4.9062.    (5)



From (4), dividing both its sides by 4, you have

    x = 1.3770 +  = 1.3770 + .

    First four positive values for n= 0, 1, 2 and 3 are (rounded to 4 decimals)  x = 1.3770,  2.9478,  4.5186,  6.0894.    (6)



Now your last step is to select three smallest numbers of 8 numbers of (5) and (6).


Doing this way, you obtain the  ANSWER  0.1938, 1.3770,  and  1.7646.