-2
$\begingroup$

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 $\pi$?

$\endgroup$
1
  • 1
    $\begingroup$ $-5892 = (-16 \times 360) + (-132).$ Then $~\displaystyle -132 \times \frac{\pi}{180} = \frac{-11}{15}\pi ~$ radians. $\endgroup$ May 16, 2022 at 12:01

2 Answers 2

1
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ does this operation have a name? like normalization? $\endgroup$
    – Alejandro
    May 16, 2022 at 12:45
  • $\begingroup$ @azerila I don't think so $-$ it's really just a $360$ degree rotation. $\endgroup$
    – user905694
    May 16, 2022 at 12:47
1
$\begingroup$

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.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged .