How do you solve:
sin(2θ) + sin(4θ) = 0
on the interval of [0 degrees, 360 degrees)
Hello,
Write : $\displaystyle \sin 4 \theta=2 \sin 2\theta \cdot \cos 2\theta$
Notice that $\displaystyle \cos A=\sqrt{1-\sin^2 A} \implies \cos 2\theta=\sqrt{1-\sin^2 2 \theta}$
Substitute : $\displaystyle y=\sin 2 \theta$
$\displaystyle \implies \sin 2 \theta+2 \sin 2 \theta \sqrt{1-\sin^2 2 \theta}=y+2y \cdot \sqrt{1-y^2}$
So you want to solve for y in $\displaystyle y+2y \cdot \sqrt{1-y^2}=0 \longleftrightarrow y(1+2\sqrt{1-y^2})=0$
Does it help ?
Note that $\displaystyle \sin(2u)=2\sin(u) \cos(u)$. Thus, $\displaystyle \sin(4\theta)=2\sin(2\theta) \cos(2\theta)$. Substituting this into the equation, we get:
$\displaystyle \sin(2\theta)+2\sin(2\theta)\cos(2\theta)=0\implie s \sin(2\theta)\left(1+2\cos(2\theta)\right)=0$
Thus we get two equations:
$\displaystyle \sin(2\theta)=0 \ \text{and} \ \cos(2\theta)=-\frac{1}{2}$. You can take it from here...