SOLUTION: Hi! I have (cos(4X))^2 + (sqrt(3)/2) * sin(4X) = 0 Anyone out there that can help me?

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Question 172642: Hi!
I have (cos(4X))^2 + (sqrt(3)/2) * sin(4X) = 0
Anyone out there that can help me?
Answer by eggsarecool(46) About Me  (Show Source):
You can put this solution on YOUR website!
%28cos%284X%29%29%5E2+%2B+%28sqrt%283%29%2F2%29+%2A+sin%284X%29+=+0
Ok so the first thing to remember is that %28cos%28x%29%29%5E2=1-%28sin%28x%29%29%5E2
which means that %28cos%284X%29%29%5E2=1-%28sin%284X%29%29%5E2
So we now rewrite your equation as 1-%28sin%284X%29%29%5E2+%2B+%28sqrt%283%29%2F2%29+%2A+sin%284X%29+=+0
which is the same as -%28sin%284X%29%29%5E2+%2B+%28sqrt%283%29%2F2%29+%2A+sin%284X%29+%2B+1+=+0
Now set y=sin%284x%29 this also means that y%5E2=%28sin%284X%29%29%5E2
Your new equation is -y%5E2+%2B+%28sqrt%283%29%2F2%29y+%2B+1+=+0
This can now be solved using the quadratic formula a=-1, b=%28sqrt%283%29%2F2%29, c=1
The formula is y+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
So you have
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ay%5E2%2Bby%2Bc=0 (in our case -1y%5E2%2B0.866025403784439y%2B1+=+0) has the following solutons:

y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%280.866025403784439%29%5E2-4%2A-1%2A1=4.75.

Discriminant d=4.75 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-0.866025403784439%2B-sqrt%28+4.75+%29%29%2F2%5Ca.

y%5B1%5D+=+%28-%280.866025403784439%29%2Bsqrt%28+4.75+%29%29%2F2%5C-1+=+-0.656712033992949
y%5B2%5D+=+%28-%280.866025403784439%29-sqrt%28+4.75+%29%29%2F2%5C-1+=+1.52273743777739

Quadratic expression -1y%5E2%2B0.866025403784439y%2B1 can be factored:
-1y%5E2%2B0.866025403784439y%2B1+=+-1%28y--0.656712033992949%29%2A%28y-1.52273743777739%29
Again, the answer is: -0.656712033992949, 1.52273743777739. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B0.866025403784439%2Ax%2B1+%29


The solver will give you the solutions to the equation
y1=-0.656712033992949
y2=1.52273743777739
But there is one more step, to make the problem clearer we set y=sin%284x%29, so we actually need a solution for x
To do this we replace they y's we got and solve for x
So we have
sin%284x%29=-0.656712033992949
sin%284x%29=1.52273743777739
Next step
4x=arcsin%28-0.656712033992949%29 use a calculator to compute this
4x=arcsin%281.52273743777739%29 when you try to evaluate this you will get an error which means that there is no solution to that part. Only the -0.656712033992949 will yield a result
Then x=%28arcsin%28-0.656712033992949%29%29%2F4 use a calculator for this two.
Enjoy :)