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Fungrim entry: a2af66

atan2 ⁣(y,x)=solution*θ(π,π][(x,y)=(rcos(θ),rsin(θ))   where r=x2+y2]\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:xRandyRand(x0ory0)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
Atan2atan2 ⁣(y,x)\operatorname{atan2}\!\left(y, x\right) Two-argument inverse tangent
UniqueSolutionsolution*xSQ(x)\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x) Unique solution
Sinsin(z)\sin(z) Sine
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Piπ\pi The constant pi (3.14...)
RRR\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(Unequal(x, 0), Unequal(y, 0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-11-11 15:50:15.016492 UTC