SOLUTION: Use inverse trigonometric functions to find the solutions of the equation that are in the interval [0, 2π).You may enter an exact answer or round your solutions to four decima

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Question 816780: Use inverse trigonometric functions to find the solutions of the equation that are in the interval [0, 2π).You may enter an exact answer or round your solutions to four decimal places (this is best for this problem). (Enter your 4 answers in a comma separated list.)
cos(x)(8cos(x) + 4) = 3
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
cos%28x%29%288cos%28x%29+%2B+4%29+=+3
The left side is factored. But the right side is not zero. So having the left side factored does not help. Multiplying out the left side:
8cos%5E2%28x%29+%2B+4cos%28x%29+=+3
Now we get a zero on the right by subtracting 3:
8cos%5E2%28x%29+%2B+4cos%28x%29+-+3=0
Now we factor, if we can. But this will not factor. It is a quadratic however. So we can use the quadratic formula with an "a" of 8, a "b" of 4 and a "c" of -3:
cos%28x%29+=+%28-%284%29%2B-sqrt%28%284%29%5E2-4%288%29%28-3%29%29%29%2F2%288%29
Simplifying...
cos%28x%29+=+%28-4%2B-sqrt%2816-4%288%29%28-3%29%29%29%2F2%288%29
cos%28x%29+=+%28-4%2B-sqrt%2816%2B96%29%29%2F16
cos%28x%29+=+%28-4%2B-sqrt%28112%29%29%2F16
cos%28x%29+=+%28-4%2B-sqrt%284%2A28%29%29%2F16
cos%28x%29+=+%28-4%2B-sqrt%284%2A4%2A7%29%29%2F16
cos%28x%29+=+%28-4%2B-sqrt%284%29%2Asqrt%284%29%2Asqrt%287%29%29%2F16
cos%28x%29+=+%28-4%2B-2%2A2%2Asqrt%287%29%29%2F16
cos%28x%29+=+%28-4%2B-4sqrt%287%29%29%2F16
cos%28x%29+=+%284%28-1%2B-sqrt%287%29%29%29%2F16
cos%28x%29+=+%28-1%2B-sqrt%287%29%29%2F4
which is short for:
cos%28x%29+=+%28-1%2Bsqrt%287%29%29%2F4 or cos%28x%29+=+%28-1-sqrt%287%29%29%2F4

Using our calculator to get decimal approximations (and rounding them to 4 places):
cos%28x%29+=+%28-1%2B2.6458%29%2F4 or cos%28x%29+=+%28-1-2.6458%29%2F4
Simplifying:
cos%28x%29+=+0.4115 or cos%28x%29+=+-0.9115

Using the inverse cos on our calculator on these decimals...
For cos%28x%29+=+0.4115, cos%5E-1%280.4115%29 is 1.1467. So the reference angle is 1.1467. And since the 0.4115 is positive and since cos is positive in the 1st and 4th quadrants we should get general solution equations of:
x+=+1.1467+%2B+2pi%2An (for the 1st quadrant)
x+=+-1.1467+%2B+2pi%2An (for the 4th quadrant)

For cos%28x%29+=+-0.9115, cos%5E-1%280.9115%29 is 0.4239. (Note: Do not enter the minus of -0.9115 when looking for a reference angle! With a minus on the decimal, cos%5E-1%280.9115%29, we will get second quadrant angle, not a reference angle!) So the reference angle is 0.4239. And since the -0.9115 is negative (here is where the minus gets used) and since cos is negative in the 2nd and 3rd quadrants we should get general solution equations of:
x+=+pi%2B0.4239+%2B+2pi%2An (for the 2nd quadrant)
x+=+pi-0.4239+%2B+2pi%2An (for the 3rd quadrant)
Replacing first pi's with 3.1416:
x+=+3.1416-0.4239+%2B+2pi%2An (for the 2nd quadrant)
x+=+3.1416%2B0.4239+%2B+2pi%2An (for the 3rd quadrant)
which simplifies to:
x+=+2.7177+%2B+2pi%2An
x+=+3.5655+%2B+2pi%2An

So the general solution equations are:
x+=+1.1467+%2B+2pi%2An
x+=+-1.1467+%2B+2pi%2An
x+=+2.7177+%2B+2pi%2An
x+=+3.5655+%2B+2pi%2An
From these we will try various integer values for n looking for x's that are in the given interval.
From x+=+1.1467+%2B+2pi%2An...
if n = 0 then x = 1.1467
if n = 1 (or larger) then x is too large for the interval
if n = -1 (or smaller) then x is too small for the interval
From x+=+-1.1467+%2B+2pi%2An...
if n = 0 (or smaller) then x is too small for the interval
if n = 1 then x = 1.1467 + 2pi = 5.1365
if n = 2 (or larger) then x is too large for the interval
From x+=+2.7177+%2B+2pi%2An...
if n = 0 then x = 2.7177
if n = 1 (or larger) then x is too large for the interval
if n = -1 (or smaller) then x is too small for the interval
From x+=+3.5655+%2B+2pi%2An...
if n = 0 then x = 3.5655
if n = 1 (or larger) then x is too large for the interval
if n = -1 (or smaller) then x is too small for the interval

So the specific solutions within the given interval are:
1.1467, 5.1365, 2.7177 and 3.5655