# Converting angles to the range $-180$ deg through $180$ deg

Is there a direct formula to transform a set of angles to $$-180$$ through $$180$$ deg? For instance, $$181$$ deg and $$541$$ deg should be translated to $$-179$$, while $$-182$$ deg and $$-542$$ deg should be translated to $$178$$. I know that for an angle $$X$$ which is not on any axis ($$0, 90, 270, 360, ...$$), when $$\text{mod}(a,b)$$ is the remainder of division of $$a$$ by $$b$$, we can write use $$k = \lfloor(\frac{\text{mod}(X,360)}{90})\rfloor$$ to determine the quadrant where angle $$X$$ resides since $$k = 0, 1, 2, 3$$ corresponds to quadrants $$1, 2, 3, 4$$ respectively. So for angle $$X$$:

$$k = 0$$ or $$1$$: $$\text{mod}(X,360)$$ gives the $$0$$ through $$180$$ range.

$$k = 2$$ or $$3$$: $$\text{mod}(X,360)-360$$ gives $$-180$$ through $$0$$ range.

Is there a way to make this simpler?

In your notation, $$\operatorname{mod}(x,360)$$ would put it in the range $$0$$ to $$360$$, and therefore, we can get what you need by $$\operatorname{mod}(x+180,360)-180$$, sending $$x$$ to the range $$-180$$ to $$180$$.

It might help to plot the graph of:

• the function you currently have,
• the function you'd like to have.

Notice that you just need to translate one graph in order to get the other.

If you want to move a graph to the left or to the right, you can adjust it with $$f(x \pm \Delta_x)$$.

If you want to move a graph up or down, you can adjust it with $$f(x) \pm \Delta_y$$.

To get the desired graph, you need to move it to the left or right by $$180°$$ (or $$540°$$, or $$900°$$, since it's periodic), and down by $$180°$$.