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I'm reading Stahl's: Geometry from Euclid to Knots. In the first part of the text, he argues that

$$\cos (A )=\frac{\cos (a)-\cos(b)\cos(c)}{\sin(b)\sin(c)}$$

and $\alpha$ an spherical angle. But by taking the $\cos \theta= \cfrac{a}{h}$ from plane geometry, and looking at:

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$enter image description here

My first guess would be to define the spherical cosine as $\cos(A)=\cfrac{b}{a}$ or $\cos(A)=\cfrac{c}{a}$ or perhaps they are defined this way in disguise and the formula for the spherical cosine actually is the suggestion I made? I guess that the definition for spherical cosine should mimic the definition for the plane cosine in some sense, but I don't see what exactly it is mimicking in this case.

Also, Stahl says that this is the cosine laws, not exactly the cosine. I looked at another pages: Wikipedia, Mathworld. And it seems that people derive the cosine laws and then, the cosine appears as a consequence. I guess in plane geometry it is the other way around but I'm not certain, at least all books I took were this way.

I have found the following here:

enter image description here

Which lines up with what I have written above but given what I wrote for $\sin$ at least, $\cos$ is still a little bit weird. But for $\sin$, I would expect to have $\sin(B)=\cfrac{b}{c}$, but I have $\sin(B)=\cfrac{\sin b}{\sin c}$ instead. Why the second instead of the first? Is it just an arbitrary decision? Is it better to do this way?

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    not a definition. A theorem. The sine and cosine are the familiar ones. The sphere has radius one. – 2017-01-17
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    @WillJagy I'm confused: Isn't the familiar ones the ones I mentioned? – 2017-01-17
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    You should check the following link. Also note that the standard law of cosine should be obtained as a limiting case of the spherical cosine law. Enjoy! [1]: http://isites.harvard.edu/fs/docs/icb.topic246919.files/March-07.pdf – 2017-01-17
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    For future visitors, I highly recommend [Spherical Trigonometry](http://www.gutenberg.org/ebooks/19770) written by I. Todhunter. Chapter IV, (page 17) onwards, including Article 37-40 provide a proof for the spherical law of cosines. Again, the law is not a definition, but a theorem that relates the sides (nomalized by the radius) to each interior spherical angle of a spherical triangle. – 2017-04-26

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