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

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) (Show Source): You can put this solution on YOUR website!
Start with the given expression

Rewrite into

Use the identity:

Use the identity: again

Use the identity:

Distribute

Factor out a sin(x)

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

So lets focus on the terms in the parenthesis
Let
So we get

Rewrite -4y into -2y-2y. This will allow us to factor

Group like terms and factor out the GCF

Factor into using difference of squares

Combine like terms (note: the common term is )

Now set each factor equal to zero. Lets start with

Now let and solve for x note: I'm using radians
and Since our interval is [0,2pi] we must ignore the negative answer. So of these two answers, is the only solution.

Now let
Distribute the 2y
Use the quadratic formula to solve for y

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=20 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 0.309016994374947, -0.809016994374947. Here's your graph:

So if we replace y with we get these solutions

or

Take the arccosine of both sides (for the solution ) to solve for x

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

Now lets use the other answer of

Take the arccosine of both sides
or Since our interval is [0,2pi] we must ignore the negative answer. So of these two answers, is the only solution.
-----------------------------------------------------------------------------
So after all of that, we find that our solutions are
(from sin(x)=0) or or or
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.
RELATED QUESTIONS
Solve for all 0 degrees less than or equal to x less than 360 degrees. 3tan x- sqrt( 3... (answered by lwsshak3)

Solve the equation for x, on the interval 0 is less than or equal to x and 2pi is greater (answered by Alan3354)

Please help me with this problem. I need to solve the equation for solutions over the... (answered by Gogonati)

find to the nearest degree, all values of x between 0 degrees and 360 degrees that... (answered by robertb)

What are the solutions from 0 to 360 degrees? Solve the following. Find all solutions (answered by rapture1965)

Please help me solve this equation: {{{ cos^2(x)+2sin(x)cos(x)-sin^2(x)=0 }}} with the... (answered by Edwin McCravy)

Find all solutions to the equation: 2sin^2x + 3sinx + 1=0?? This is how I attempted to... (answered by Alan3354)

find all the angles x between 0 degrees and 360 such that 5 sin... (answered by solver91311)

Find all solutions for, 5 sin^2(x) plus sin(x)= cos^2(x), that lie between 0 =< x =<... (answered by lwsshak3)