Question: I'm working through Example 2.22 in Steward and Tall's Algebraic Number Theory book. The goal is to determine the ring of integers of $\mathbb Q(\sqrt[3]5)$. Let $\theta\in\mathbb R$ such that $\theta^3=5$. Let $\omega=e^{2\pi i/3}$. I'm at the point where I need to check whether
$$
\alpha=\frac 13(1+\theta+\theta^2)
$$
or
$$
\beta=\frac 13(2+2\theta+2\theta^2)
$$
are algebraic integers (I've checked the other cases). Let's start with $\alpha$. I can think of two ways to do this. The first would be to consider the norm
$$
N(\alpha)=\frac 1{27}(1+\theta+\theta^2)(1+\omega\theta+\omega^2\theta^2)(1+\omega^2\theta+\omega\theta^2).
$$
If $\alpha$ is an algebaric integer, then $N(\alpha)\in\mathbb Z$. However, the expression of $N(\alpha)$ is a bit complicated (unless I'm overlooking something). The second would be to determine the minimum polynomial $\mu$ of $\alpha$ over $\mathbb Q$, and then invoke that $\alpha$ is an algebraic integer iff $\mu\in\mathbb Z[X]$.

In both cases I'm not too hopeful. Should I still proceed in this way, or is there something obvious I'm missing?

Answer: 
[![enter image description here](https://i.stack.imgur.com/7hWo2.png)](https://i.stack.imgur.com/7hWo2.png)

As pointed out by dezdichado, the problem can be done by two applications of Cosine Formula.

Refer to the figure,

One such formula is:

$$\cos \alpha=\frac{(c-x)^2+y^2-a^2}{2(c-x)y}$$

Applying Cosine Formula to the other triangle and combining the $2$ equations would yield the required answer.
I have an idea to start :

In $\Bbb C, A=0, B=1, C=\xi +i\eta, D=x$

Unfortunately, none of my calculations succeed, although simplifications appear: I suspect that there is a sign error in the proposed equality.