Question #6192d

Jun 2, 2017

$\theta = \pi \pm \pi n$; $n \in \mathbb{Z}$

Explanation:

I'm assuming you mean to solve ${\left({\sin}^{2} \left(\theta\right)\right)}^{2} = 0$.

$R i g h t a r r o w {\left({\sin}^{2} \left(\theta\right)\right)}^{2} = 0$

$R i g h t a r r o w {\sin}^{2} \left(\theta\right) = 0$

$R i g h t a r r o w \sin \left(\theta\right) = 0$

Now, let's take a look at the graph of $y = \sin \left(\theta\right)$:

graph{sin(x) [-10, 10, -5, 5]}

The $x$-axis is the axis for all values of $\theta$.

We are looking for the values of $\theta$ for which $\sin \left(\theta\right) = 0$.

On the graph, we can see that $y = 0$ for multiple values of $\theta$, so there will be multiple solutions to the equation $\sin \left(\theta\right) = 0$.

Also, you haven't specified a domain, so the solutions can be positive or negative. The coordinates of some of the solutions are shown in the graph above.

The solutions are all multiples of $\pi$, i.e. $\theta = \pi \pm \pi n$ for some integer $n$.

Let's express the solutions formally:

$\therefore \theta = \pi \pm \pi n$; $n \in \mathbb{Z}$