# Goniometric equation

1. Mar 3, 2013

### scientifico

1. The problem statement, all variables and given/known data
sen(2x) * sen(x) = sen(4x) * sen(3x)

3. The attempt at a solution
I applied product to sum and sum to product identities and now I get cos(3x)=cos(7x ) how can I solve it?

thank you

2. Mar 3, 2013

### tiny-tim

hi scientifico!
yup!

so 3x = ±7x + … ?

(alternatively, use the formula for cosA - cosB)

3. Mar 3, 2013

### scientifico

What is this?

4. Mar 3, 2013

### tiny-tim

draw the graph of cosθ …

what has to be the relation between θ1 and θ2 if cosθ1 = cosθ2 ?

5. Mar 3, 2013

θ1 = θ2 ?

6. Mar 3, 2013

### tiny-tim

yes, but what are all the other solutions?

(and have you drawn a graph of cosθ ?)

7. Mar 4, 2013

### scientifico

Yes but i don't know what to search for

8. Mar 4, 2013

### SammyS

Staff Emeritus
If cos(3x) = cos(7x), then you have cos(7x) - cos(3x) = 0 .

Use the sum to product identities to change this to the product of two sines .

$\displaystyle \cos \theta - \cos \varphi = -2\sin\left( {\theta + \varphi \over 2}\right) \sin\left({\theta - \varphi \over 2}\right)$

It's almost always easier to solve an equation with product that equals zero than one with a sum/difference that equals zero.

9. Mar 5, 2013

### scientifico

ok but that way I get cos(5x)

10. Mar 5, 2013

### tiny-tim

no, you get 2sin(5x)sin(2x) = 0