Solving Goniometric Equation: sen(2x) * sen(x)=sen(4x)*sen(3x)

[scientifico]

Homework Statement

sen(2x) * sen(x) = sen(4x) * sen(3x)

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

 

[tiny-tim]

hi scientifico!

I applied product to sum and sum to product identities and now I get cos(3x)=cos(7x)

yup!

so 3x = ±7x + … ?

(alternatively, use the formula for cosA - cosB)

 

[scientifico]

so 3x = ±7x + … ?

What is this?

 

[tiny-tim]

draw the graph of cosθ …

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

 

[scientifico]

θ1 = θ2 ?

 

[tiny-tim]

yes, but what are all the other solutions?

(and have you drawn a graph of cosθ ?)

 

[scientifico]

Yes but i don't know what to search for

 

[SammyS]

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.

 

[scientifico]

ok but that way I get cos(5x)

 

[tiny-tim]

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