Fungrim entry: a2af66

\operatorname{atan2}\!\left(y, x\right) = \mathop{\operatorname{solution*}\,}\limits_{\theta \in \left(-\pi, \pi\right]} \left[\left(x, y\right) = \left(r \cos(\theta), r \sin(\theta)\right)\; \text{ where } r = \sqrt{{x}^{2} + {y}^{2}}\right]

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)

TeX:

\operatorname{atan2}\!\left(y, x\right) = \mathop{\operatorname{solution*}\,}\limits_{\theta \in \left(-\pi, \pi\right]} \left[\left(x, y\right) = \left(r \cos(\theta), r \sin(\theta)\right)\; \text{ where } r = \sqrt{{x}^{2} + {y}^{2}}\right]

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`Atan2`	$\operatorname{atan2}\!\left(y, x\right)$	Two-argument inverse tangent
`UniqueSolution`	$\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x)$	Unique solution
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Pi`	$\pi$	The constant pi (3.14...)
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("a2af66"),
    Formula(Equal(Atan2(y, x), UniqueSolution(Brackets(Where(Equal(Tuple(x, y), Tuple(Mul(r, Cos(theta)), Mul(r, Sin(theta)))), Equal(r, Sqrt(Add(Pow(x, 2), Pow(y, 2)))))), ForElement(theta, OpenClosedInterval(Neg(Pi), Pi))))),
    Variables(x, y),
    Assumptions(And(Element(x, RR), Element(y, RR), Or(NotEqual(x, 0), NotEqual(y, 0)))))

Topics using this entry

Inverse tangent