# Sin, Cos, then?

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?

• – Mauro ALLEGRANZA Jan 30 at 14:49
• Check out "solid trigonometry": mathworld.wolfram.com/SolidAngle.html – user247327 Jan 30 at 14:53
• 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. – Mark Bennet Jan 30 at 15:07

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.