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I hope this random thought is on topic. If not please notify me before penalising me for it.

I have reflected that Sin and Cos provide answers for trigonometry problems with cartesian pairs on a 2D plane inside a unit circle.

Why stop there?

Is there a further similar function which adds the z axis so problems can be solved in 3D, inside a unit sphere? If there isn't, is there an explanation for why?

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    $\begingroup$ See Spherical trigonometry. $\endgroup$ – Mauro ALLEGRANZA Jan 30 at 14:49
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    $\begingroup$ Check out "solid trigonometry": mathworld.wolfram.com/SolidAngle.html $\endgroup$ – user247327 Jan 30 at 14:53
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    $\begingroup$ The history of the move from two dimensions to three is an interesting one and led Hamilton to discover the Quaternions. Also observe that rotations of a sphere are possible with an infinite number of axes and that the central involution (reflection in the centre) has determinant $-1$ in three dimensions. None of this is precisely on your point, but it shows some issues which arise in finding the most convenient way to work with a sphere in contrast to a circle. $\endgroup$ – Mark Bennet Jan 30 at 15:07
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In spherical co-ordinates each point in $\mathbb{R}^3$ has a distance from the origin $r$ and two angles $\theta$ and $\phi$. This is analogous to polar co-ordinates in $\mathbb{R}^2$ except there are two angles instead of one.

Converting to Cartesian co-ordinates, the $x,y,z$ co-ordinates of a point are functions of $r$ and the two angles:

$(x,y,z) = \left(rf(\theta, \phi), \space rg(\theta, \phi), \space rh(\theta, \phi) \right)$

but, as it happens, the functions $f,g,h$ can be expressed simply in terms of $\sin$ and $\cos$:

$f(\theta, \phi)=\sin (\theta) \cos (\phi) \\ g(\theta, \phi)=\sin (\theta) \sin (\phi)\\ h(\theta, \phi)=\cos (\theta)$

As far as I know the functions $f, g, h$ have never been given specific names.

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