# Intro - Spheres?? *You should probably finish ellipses and conics first.* [[Conics - Circles, Parabolas & More! (Maths)|Get back to reality and click this. ]] It's the 3D circle, and we'll be doing trigonometry on it. Not sure what else you expected... This course is typically expected of *second-third year* physics or maths majors. Even then, a good lot of people don't take this course until they reach the graduate level. I hope this gives you some faith in yourself, because if the [IOAA](https://www.ioaastrophysics.org) asks you to do it, anyone can! Good luck! Here, you'll have to think three-dimensionally, which is easier said than done. Goodbye, 2D thought process! Hello, personal enlightenment trying to understand half of this at three in the morning. Also, a 'triangle' doesn't have to be 180 degrees on a sphere. I'm not joking. Have fun! %%put a funky image here :)%% # Spherical Trigonometry Terms and Conditions Every branch of trigonometry has its laws. Some are neat mathematical tricks, others are word-of-mouth definitions that allow the discipline to hold true. The same can be said with spherical trigonometry - only that it's a little worse than all the others. Let's take a look at some of the 'rules'! >[!Tip] Useful Definitions >- **Great Circle:** A circle along the sphere with the same length as its equator. For example, a star moving across our night sky has an orbit of a great circle! > >- **Minor Circle:** A circle on our sphere with a circumference *shorter* than the great circle. An example of this is a circumpolar orbit (link to the celestial coords thing), which is why the sun doesn't set sometimes in the winter. # The Good - Spherical Sine and Cosine Rules Some of the first trigonometrical laws we learn are the sine (link) and cosine (link) rules. While considered by many to be menial and repetitive in their usage, being able to identify when and where they can be used is a very, very useful skill that's commonly invoked. On spheres, the situation's a little different. For now, let's consider a sphere with a great circle around it. Have a handy diagram! ![[Pasted image 20240216205713.png]] Now let's draw a couple more to make a pseudo-triangle. ![[Pasted image 20240216210054.png]] There. Notice the part in the middle. Like any 'triangle', it's wise to assign each side and corresponding angle its own variable, just for simplicity's sake. ![[Pasted image 20240216210339.png]] And here's where you've fallen into my trap. See, the lower case variables, the ones that pertain to the arcs - they're all actually the *angle of the great circle arc* - so a part of 360 degrees. This let's us write our first rule, the **spherical sine rule!** $\frac{\sin c}{\sin C} = \frac{\sin b}{\sin B} = \frac{\sin a}{\sin A}$ The cosine law also applies to this - though it looks foreign compared to the sine rule. $\cos c = \cos a\cos b +\sin b\sin c\sin C$ There's another way of writing it - but that's for later. Derivations down below, and I'd highly recommend checking them out (once they're done, of course.) >[!Abstract]- Proving the Cosine Rule >Before we begin, let's draw a diagram. > >![[Pasted image 20240405234925.png]] > >For our case, angle OAD and OAE are 90 degree angles. The arclengths $b$ and $c$ (in radians) equal to the separation are equal to DAO and EOA respectively. Therefore, we can write: >$\overrightarrow{OD} = OA\sec b$ >$\overrightarrow{AD} = OA\tan b$ >$\overrightarrow{OE} = OA\sec c$ >$\overrightarrow{AE} = OA\tan c$ >Note how there are two triangles on the diagram - ODE and ADE. We can find two expressions for $\overrightarrow{DE}$ by taking $a$ as the total angular separation of the system: >$\overrightarrow{DE}^2 = OA^2\sec^2 b + OA^2\sec^2 c - 2(OA\sec b \ OA\sec c)\cos A = OA^2\tan^2 b + OA^2\tan^2 c - 2(OA\tan b OA\tan c)\cos A$ >So equate both expressions and resolve using the trig identity $sec^2 x - 1 = tan^2x$. That gives you: >$\cos a = \cos b\cos c-\sin c\sin b\cos A$