Trig equation: sin(5x)+sin(x)=0

Question: Solve for .

I know ways to solve this by expanding sin(5x) but I'm wondering if the method I'm attempting below is correct and how to proceed with it:

where and are integers.

Now the solution interval is between 0 and pi so we have the following inequality:

There are infinite integer solutions to this inequality so how do I limit the solutions further?

Re: Trig equation: sin(5x)+sin(x)=0

Quote:

Originally Posted by

**Josh146** Question: Solve

for

.

I know ways to solve this by expanding sin(5x) but I'm wondering if the method I'm attempting below is correct and how to proceed with it:

**<--- hmmmm, I never heard about that identity**
...

Here is how I would have done this question:

1.

2. That means your equation becomes:

3. Factor out 2sin(x):

4. A product equals zero if one of the factors equals zero:

5. For the 2nd equation use the substitution . So you have to solve for u:

which yields 2 positive solutions. Use them to determine x.

Re: Trig equation: sin(5x)+sin(x)=0

Re: Trig equation: sin(5x)+sin(x)=0

Thank you very much for both of the replies but I know these methods.

The reason I set up the thread is to check if the method I gave a) is correct so far and b) can be used to find all the solutions.

Quote:

Originally Posted by

**earboth** **hmmmm, I never heard about that identity**

The identity I used was due to the fact that sin(x) is an odd function.

Re: Trig equation: sin(5x)+sin(x)=0

I realised that my method would miss some solutions. Instead it should be:

And this will make things too confusing when the inequalities are produced. I think I'll give up on this now.

Re: Trig equation: sin(5x)+sin(x)=0

Actually there's a much simpler way to do it:

or

where and are integers.

Subbing in and leads to the solutions much faster than using the methods given by other people.

Re: Trig equation: sin(5x)+sin(x)=0

The simplest way I can think of to solve this is to use the general solution for trigonometric equations, which in this case is

If

then

This gives you all possible solutions for for integer .

So, the problem looks like this:

If you now start plugging in values for from 0 onwards you should end up with the following valid answers (obviously restricted by the range you are given for ).:

I hope that's helpful.

Re: Trig equation: sin(5x)+sin(x)=0

THere's the trig identity

,

so the equation can be rewritten as

so either or