Determine the ranges for sums and differences of two angles

Question

Suppose that $\alpha \in [\alpha_1,\alpha_2)$ and $\beta \in [\beta_1,\beta_2)$ are two angles. In this case, what will be the ranges of $\alpha+\beta$ and $\alpha-\beta$?

Answer 1

$\alpha_1 \le \alpha < \alpha_2$ and $\beta_1 \leq \beta \lt \beta_2$. From the range of $\beta$ we get that $-\beta_1 \geq -\beta >-\beta_2$.

Adding the first two inequalitis gives $\alpha_1+\beta_1 \leq \alpha+\beta < \alpha_2+\beta_2$, or $\alpha+\beta\in [\alpha_1+\beta_1, \alpha_2+\beta_2)$.

Similarly, the first and third inequalities imply $\alpha-\beta \in (\alpha_1-\beta_2, \alpha_2-\beta_1)$.