1. ## trigonometry

sin6x+3sin4x.cos2x=0
find solution set

sin6x+3/2(sin6x+sin2x)=0
5sin6x+3sin2x=0

I got stuck here.

2. ## Re: trigonometry

Originally Posted by kastamonu
sin6x+3sin4x.cos2x=0
find solution set

sin6x+3/2(sin6x+sin2x)=0
5sin6x+3sin2x=0

I got stuck here.
Is it $\displaystyle sin(2x)$ or $\displaystyle sin^2(x)$?

-Dan

3. ## Re: trigonometry

write

sin 6x = sin( 4x + 2x) = ......

sin2x

5. ## Re: trigonometry

Originally Posted by Idea
write

sin 6x = sin( 4x + 2x) = ......
I couldn't get anything.

6. ## Re: trigonometry

$\displaystyle \sin (4x+2x)=\sin 4x \cos 2x +\cos 4x \sin 2x$

7. ## Re: trigonometry

Let y = 2x so we have 5.sin(3y) + 3.sin(y) = 0

where sin(3y) = sin(2y+y) = sin(2y).cos(y) + cos(2y).sin(y)

= 2.sin(y).cos^2(y) + sin(y).(1-2.sin^2(y))

= 2.sin(y)(1-sin^2(y)) + sin(y) - 2.sin^3(y)

= 3.sin(y) - 4.sin^3(y)

8. ## Re: trigonometry

Plug that in for sin(3y). Then, let $\displaystyle u = \sin(y)$. You wind up with a cubic equation in $\displaystyle u$. Solve for $\displaystyle u$ (you have something for the form $\displaystyle au-bu^3=0$, so $\displaystyle u(a-bu^2)=0$ implies $\displaystyle u=0$ or $\displaystyle a-bu^2=0$). Once you solve for $\displaystyle u$, you have $\displaystyle u = \sin(y)$, so take arcsin of both sides. Then, you have $\displaystyle y = 2x$.

9. ## Re: trigonometry

you could also linearize the equation

$\displaystyle \sin 2x (4+ 5\cos 4x)=0$

giving two sets of solutions

$\displaystyle \sin 2x=0$

$\displaystyle \cos 4x=-4/5$

Thanks.