# Thread: How to approach general solutions?

1. ## How to approach general solutions?

If I want to find the general solution of

$sin2\theta + sin4\theta = cos\theta$ how would I go about doing this?

What would be a possible method that you could use to solve this question

Thanks if you can help me

2. Originally Posted by db5vry
If I want to find the general solution of

$sin2\theta + sin4\theta = cos\theta$ how would I go about doing this?

What would be a possible method that you could use to solve this question

Thanks if you can help me
break things down to sine and cosine. use your trig identities.

you have: $2 \sin \theta \cos \theta + 2 \sin 2 \theta \cos 2 \theta = \cos \theta$

$\Rightarrow 2 \sin \theta \cos \theta + 4 \sin \theta \cos \theta \cos 2 \theta = \cos \theta$

Note that if $\cos \theta = 0$ we have a solution. So find all the $\theta = \frac {n \pi}2$ where $n$ is any odd integer is one set of solutions.

Now assume $\cos \theta \ne 0$, we can divide by it to get

$2 \sin \theta + 4 \sin \theta \cos 2 \theta = 1$

Now change out $\cos 2 \theta$ to something more convenient and continue

(instead of this thing about considering whether cosine is zero or not, the standard way would be to bring the cosine on the right to the left, factor it out and set each factor to zero)