SOLUTION: solve 7cos(4x)=5 for the smallest three positive solutions. Give answer accurate to at least two decimal places as a list separated by commas. can someone please help me better u

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Question 1183233: solve 7cos(4x)=5 for the smallest three positive solutions.
Give answer accurate to at least two decimal places as a list separated by commas.
can someone please help me better understand how to do this problem please
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

7cos%284x%29=5 for the smallest three positive solutions.
cos%284x%29=5%2F7+
4x=cos%5E-1%285%2F7+%29
x=cos%5E-1%285%2F7+%29%2F4
in general:
x=cos%5E-1%285%2F7+%29%2F4%2B%28pi%2F2%29%2An
x=pi%2F2-cos%5E-1%285%2F7+%29%2F4%2B%28pi%2F2%29%2An

three smallest exact solutions:
x=+cos%5E-1%285%2F7+%29%2F4%2B%28pi%2F2%29%2A0=>x=cos%5E-1%285%2F7+%29%2F4
x=cos%5E-1%285%2F7+%29%2F4%2B%28pi%2F2%29%2A1=>x=cos%5E-1%285%2F7+%29%2F4%2Bpi%2F2
x=pi%2F2-cos%5E-1%285%2F7+%29%2F4%2B%28pi%2F2%29%2A0=>x=pi%2F2-cos%5E-1%285%2F7+%29%2F4+

approximate solutions in decimal form:
x=0.1937975
x=1.3769988267948965
x=1.7645938267948966

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
solve 7cos(4x)=5 for the smallest three positive solutions.
Give answer accurate to at least two decimal places as a list separated by commas.
can someone please help me better understand how to do this problem please
~~~~~~~~~~~~~ 


Your starting equation is 

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


As you understand, it is EQUIVALENT to

    cos(4x) = 5%2F7.       (2)


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

    4x = arccos%285%2F7%29 + 2pi%2An = 0.775193 + 2pi%2An                        (3)

and 

    4x = 2pi+-+arccos%285%2F7%29 + 2pi%2An = 2%2A3.14159-0.775193 + 2pi%2An,     (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 + %28pi%2F2%29%2An = 0.193798 + %283.14159%2F2%29%2An.

    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 + %28pi%2F2%29%2An = 1.3770 + %283.14159%2F2%29%2An.

    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.

Solved.