SOLUTION: find all solutions for equation with value of x between 0 and 360 degrees: 1. sin(4x) + sin x = 0 I have 2(2sin(x) cos (x))(cox(2x)) + sin x = 0 so far but don't know which ide

Algebra ->  Trigonometry-basics -> SOLUTION: find all solutions for equation with value of x between 0 and 360 degrees: 1. sin(4x) + sin x = 0 I have 2(2sin(x) cos (x))(cox(2x)) + sin x = 0 so far but don't know which ide      Log On


   

Question 77922: find all solutions for equation with value of x between 0 and 360 degrees:
1. sin(4x) + sin x = 0
I have 2(2sin(x) cos (x))(cox(2x)) + sin x = 0
so far but don't know which identity to use for cos(2x) that will help solve.
This is from my daughter's textbook but I don't have the book with me now. She's taking pre-calc in high school. Thanks
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
sin%284x%29+%2B+sin+x+=+0 Start with the given expression

sin%282%2A%282x%29%29+%2B+sin+x+=+0 Rewrite sin%284x%29 into sin%282%2A%282x%29%29

2sin%282x%29cos%282x%29+%2B+sin+x+=+0 Use the identity: sin2x=2sinxcosx

2%282sin%28x%29cos%28x%29%29cos%282x%29+%2B+sin+x+=+0 Use the identity: sin2x=2sinxcosx again

%284sin%28x%29cos%28x%29%29%282%28cos%28x%29%29%5E2-1%29+%2B+sin+x+=+0 Use the identity: cos2x=2%28cos%28x%29%29%5E2-1


8sin%28x%29%28cos%28x%29%29%5E3-4sin%28x%29cos%28x%29+%2B+sin+x+=+0 Distribute 4sin%28x%29cos%28x%29

sin%28x%29%288%28cos%28x%29%29%5E3-4cos%28x%29+%2B+1%29+=+0 Factor out a sin(x)

Since we know the value of x for sin%28x%29=0 (the solution is x=0) we can ignore the sin(x) and try to solve the expression in the parenthesis

8%28cos%28x%29%29%5E3-4cos%28x%29+%2B+1+=+0 So lets focus on the terms in the parenthesis
Let y=cos%28x%29
So we get
8y%5E3-4y%2B1=0

8y%5E3-2y-2y%2B1=0 Rewrite -4y into -2y-2y. This will allow us to factor

2y%284y%5E2-1%29-%282y-1%29=0 Group like terms and factor out the GCF

2y%282y%2B1%29%282y-1%29-%282y-1%29=0 Factor 4y%5E2-1 into %282y%2B1%29%282y-1%29 using difference of squares


%282y%282y%2B1%29-1%29%282y-1%29=0 Combine like terms (note: the common term is 2y-1)

Now set each factor equal to zero. Lets start with 2y-1
2y-1=0
2y=1
y=1%2F2
Now let cos%28x%29=1%2F2 and solve for x note: I'm using radians
x=1.04719%2Bpi%2An and x=-1.04719%2Bpi%2An Since our interval is [0,2pi] we must ignore the negative answer. So of these two answers, x=1.04719%2Bpi%2An is the only solution.




Now let 2y%282y%2B1%29-1=0
4y%5E2%2B2y-1=0 Distribute the 2y
Use the quadratic formula to solve for y
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ay%5E2%2Bby%2Bc=0 (in our case 4y%5E2%2B2y%2B-1+=+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=%282%29%5E2-4%2A4%2A-1=20.

Discriminant d=20 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+20+%29%29%2F2%5Ca.

y%5B1%5D+=+%28-%282%29%2Bsqrt%28+20+%29%29%2F2%5C4+=+0.309016994374947
y%5B2%5D+=+%28-%282%29-sqrt%28+20+%29%29%2F2%5C4+=+-0.809016994374947

Quadratic expression 4y%5E2%2B2y%2B-1 can be factored:
4y%5E2%2B2y%2B-1+=+4%28y-0.309016994374947%29%2A%28y--0.809016994374947%29
Again, the answer is: 0.309016994374947, -0.809016994374947. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B2%2Ax%2B-1+%29


So if we replace y with cos%28x%29 we get these solutions

cos%28x%29=0.309017 or cos%28x%29=-0.809017

Take the arccosine of both sides (for the solution cos%28x%29=0.309017) to solve for x

x=1.25664%2Bpi%2An or x=-1.25664%2Bpi%2An Here are 2 more possible solutions. Since our interval is [0,2pi] we must ignore the negative answer. So of these two answers, x=1.25664%2Bpi%2An is the only solution.

Now lets use the other answer of cos%28x%29=-0.809017

cos%28x%29=-0.809017
Take the arccosine of both sides
x=2.51327%2Bpi%2An or x=-2.51327%2Bpi%2An Since our interval is [0,2pi] we must ignore the negative answer. So of these two answers, x=2.51327%2Bpi%2An is the only solution.
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So after all of that, we find that our solutions are
x=0 (from sin(x)=0) or x=1.04719%2Bpi%2An or x=1.25664%2Bpi%2An or x=2.51327%2Bpi%2An
As always, we can check our work by using a calculator.
This is a lot to take in, so feel free to ask me further about any of this.