# How to find an equivalent angle between $-\pi$ and $\pi$? [closed]

Let's say we have an angle such as $$270$$ degrees or $$-5892$$ degrees, or similarly in radians.

How do we convert it to its equivalent value between $$-\pi$$ and $$\pi$$?

• $-5892 = (-16 \times 360) + (-132).$ Then $~\displaystyle -132 \times \frac{\pi}{180} = \frac{-11}{15}\pi ~$ radians. May 16, 2022 at 12:01

Given $$\theta$$, you can find an equivalent angle $$\alpha = \theta + 2k\pi, \; k \in \mathbb{Z} \;$$ s.t. $$\;-\pi \le \alpha \le \pi$$.
$$k = \left\lfloor \frac{\pi - \theta}{2\pi} \right\rfloor$$ where $$\lfloor \cdot \rfloor$$ is the floor function.
Note: If $$\theta$$ is in degrees, it helps to convert it to radians first.
• @azerila I don't think so $-$ it's really just a $360$ degree rotation.
It helps to express the angle in radians first. $$1$$ radian is equal to $$\frac{180}{\pi}$$ degrees. Equivalently, 1 degree is equal to $$\frac{\pi}{180}$$ radians. Use this and express the given angle in radians. Next, recognise that the complete angle around a point is equal to $$2\pi$$ radians. Therefore, $$2\pi+1$$ radians is equivalent to $$1$$ radian. If the angle you so get is greater than or equal to $$\pi$$ radians, then you can express it as a negative angle in $$[-\pi,0]$$. Say you get $$x$$ radians as the answer. The same angle expressed as a negative angle will be $$-(2\pi-x)$$radians.