Classical spherical trigonometry | Universal Hyperbolic Geometry 36 | NJ Wildberger

This video presents a summary of classical spherical trigonometry. First we define spherical distance between two points on a sphere, then the angle between two lines on a sphere (i.e. great circles). After a quick reminder of the circular functions cos,sin and tan, we present the main laws: the (spherical) sine law, two Cosine laws, Pythagoras' theorem, Thales theorem, Napier's Rules, the law of haversines and a few more, namely Napier's and Delambre's Analogies.
We will not be using these laws in any substantial way: shortly in this course we will redevelop the framework of spherical trigonometry in a completely new, simplified algebra fashion. We will obtain analogs of the above laws, but analogs that are much more rigorous, powerful and accurate.
Video Content:
00:00 Introduction
2:35 Angle between lines
7:06 Notation for a spherical triangle
11:07 Circular functions
18:21 Spherical sine law
21:17 Pythagoras 1
25:35 The law of haversines
29:38 Half angle formulas
30:57 Napier's Analogies
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[@btpcmsag]

What a RELIEF it is to find your videos, Professor Wildberger! Your pace, foresight, cogency, diction, precision, thoughtfulness, confidence, patience and clarity are unsurpassed among all the trig presentations I have seen for over 60 years. And I'm still only on my first video of yours. Tally-ho!!

[@arnabmukherjee5840]

Made my day. I am an Amateur astronomer. It's a holy grail man. Love you.

[@carmelpule8493]

Congratulations. This is a wonderful introduction to the concept of spherical geometry which is related to rotations. My Grandson is now coming of age. and I thought about introducing him as to how a linear measure can be processed to form a periodic function when a linear measure is rotated around , say a cylinder. So a long piece of string will really never end and one which will reach a great distance , but if the long piece od string is wrapped around a cylinder then there will be periods in the sting which will repeating the same trajectory or superimposed circles, If one views the wrapping function on the cylinder from a long way away and from position in quadrature space locations then the sine and cosine function will be seen rather than seeing them as a time phase function separate by pi/2 To add a concept that all this is so natural as the relation between the circumference and the diameter which is pi= 3.142 which is such a natural ration, I though I would uses the other natural number of growth and decay and rotation which is the exponential function " e " = 2.718. which is a growth or a decay funtion with time if it is written as e^t. This is a growth or a decay which can be expressed by normal numbers, If "e" is now processed as e^j.t where j is a 90 degree operator this will lead to e^j(t)= cos(t) +j.sin(t) If jt is substituted in all the series expansion of sinx and cos x shown at 14:35 then the even powers will remain real numbers while the odd powers will be in quadrature and so they will be added as Vectors . Adding those quadrature vectors in the series will produce a rotation action od e^jt and it is wonderful seeing them targeting on a circumference of a circle eventually like growing Fractals. The fact that in spherical triangles the sides subtends an angle at the centre of the sphere they the sides do in fact represent the centre angles in radians, This is the beauty of your lesson, you explained that very well at the start, Since the sides are also representing angles then all the spherical formulas become related to the flat formulas relating the sides and angles of a flat triangle or tirangles drawn on a plane, ' Congratulations for your presentation. My Grandson will soon be introduced to all this, Well done and thank you.

[@conanladler3472]

I just recently got interested in spherical trig this is honestly just want the best trig course I have ever had

[@popytkpisatel]

Very clear and understandable way it's explained. Thank you for this content

[@ceyhunkarasu8925]

Thank you for this video. It is just what I was looking for and it is explained in a very comprehensive manner.

[@MarkLares]

Thank you so much for your video, i really appreciate you taking the time to do this.

[@njwildberger]

You're welcome!

[@brianb4390]

Thirty-six seconds in, and he is already waving a drumstick around. I have a feeling this video is going to Rock !!!

[@sudaama108]

Well presented overview of this topic...you are an excellent teacher, able to simplify complex ideas :-)

[@pennyoflaherty1345]

Keep up with Your Great efforts with teaching others--We gradually glean these concepts & the penny drops. Many thanks again for Your Comprehensive Coverage .T O'Flaherty

[@revelationreflection]

Great teacher. Thank you very much!

[@maramarosifavago]

You do an AWESOME work ! Thank you !!!!!! :)

[@hexenpl]

thanks... a lot ;) that's what i was looking for. Thumbs up

[@shahrammakoui859]

What a wonderful presentation, as always

[@anisrazlin2951]

sir njwildberger, thanks for this super brilliant video!

[@abdulhalim520]

You're the greatest njwildberger. Could you please do videos on: • Differentiation II: stationary points, maximum, minimum and points of inflection • Differentiation III: compound functions such as function of a function, product and quotient • Theory of quadratic equations: roots and discriminants • Logarithms: laws and their simple manipulations • Trigonometry (compound angles 1): finding sin, cos or tan of a compound angle, simplifying and solving trig equations • Trigonometry (compound angles 2): expressing A cos x + B cos x in a single trigonometric ratio, solving A cos x + B sin x = constant, equations involving double angles • Integration: definite and indefinite integration • Integration: areas and volumes • Differentiation of trigonometric functions It would really help me as a student to understand all these topics. Also, • Differentiation of Trigonometric Functions • Integration of Trig Functions • More Trig Integration • Functions and Inverse Functions • Logarithmic and Exponential functions • Integration by Substitution • Integration by Parts • Polynomials and the Factor Theorem It would help me a lot, and I am also happy to donate!

[@njwildberger]

Yes it's a prelude from J.S. Bach's Well-Tempered Clavier.

[@GeodesicBruh]

njwildberger eheheheh I could tell, thank you :)

[@joanzapanta5940]

Effort much appreciated 😁

[@krishnasrinivasan7541]

Thank you very much! Very nice video.

[@colinkiama]

Thanks for the video!

[@abirislam95]

Thanks sir. I learn a lot of names in this class.

[@mohammadtanvirrahman5541]

Very easy and simply explained.

[@shivanisorte7088]

Thank you very much, sir.

[@meleknurgorgun4652]

Thank you for the video sır 🙌 it helped me a lot when I was studying spherical astronomy class🎆

[@Alrescha]

Thank you, props ball has inspired me

[@gototcm]

Good stuff. Can you provide a reference for the derivation of some of these spherical formulas?

[@PhysicsBanglaTutorial]

Nice

[@elvinceleonelle6378]

Thank sir u really help me so much

[@sarisap.5429]

Thank you so much

[@anjanapdas9060]

thank you very much. I have a doubt, How can we find out the parameters A, B, C and a,b,c from a spherical triangle formed by the overlapping of 3 spheres.Given the center and radius of 3 spheres.

[@paoloperez7792]

Thank you for making this video because ill use it in my engineering course

[@njwildberger]

Paolo Perez You are welcome.

[@Kampup]

njwildberger Hi, I tried to piece this together in my own way. can you help me with this equation? NASA tell's us that the earth is 25 thousand miles in circumference. you can find an 8 inch drop after the first mile. After the second mile 2 sqrd 4 x 8=32 inches drop. After the 3rd mile 3x3 =9 x8=72 so 3 miles out we should be 72 inches below sea level, is this correct? this gives the earth a curvature of 8 inches per mile squared is this correct? On a sphere of 25,000 miles in circumference using pythagorean theorem is this the dip we can expect?

[@paoloperez7792]

njwildberger sir do you have lectures on spherical trigonometry?

[@mechmania8450]

What would be the ration between the angular difference A-B and B-C and the great circle distances in the unit radius sphere as in measuring the distances between 2 stars in the sky. Also the ancients worked out sines by geometry Kepler used these.

[@alecjohnbriones9299]

Thank you very much!!!! :))

[@hyperduality2838]

Sine is dual to cosine -- perpendicularity in hyperbolic geometry is measured in terms of duality!

[@jdtaramona1]

Hi Dr. Wildberger, first of all is a great effort that you providing very insightful comments and teaching about hyperbolic trigonomentry, particullarly spherical. I want to ask you a question about if is possible to derivate the so called Law of Haversines starting from the cross product of two vectors (P1 and P2)? Thank you in advance and please, go on with your classes. I am also interested in Elliptic Integrals.

[@ramseylloydmartin9996]

thanks

[@michael_toms]

Would this be the work of Jacques Cassini, a French astronomer, and son of the famous Italian astronomer Giovanni Domenico Cassini?

[@skhalid360]

Mr Wildberger, I am very confused. You stated these theorems, particularly the spherical Pythagorean theorem, and I became excited and started calculating distances on the Earth, using the latitudes and longitudes of various cities. I used the mean radius of the Earth. But when I calculated my distances, my answers were nowhere near what GPS told me. Is the Earth not a good enough sphere?

[@manojkumarpoovakulamsebast9648]

oblate spheroid

[@billyzig]

Sohaib Khalid Neil Degrasse Tyson said its pear shaped, like his head

[@greg12145]

Maybe you can post your results... I'd love to look at them... I've been using formulas from the Washington State Surveying site and they don't make sense either with larger values. Or, what I'm trying to accomplish requires different methodology... But, he did state that classical/standard methods are shady.

[@danodet]

Recently I realised that the angle between two planes is not a max, not a min but a saddle point.

[@karimkhan1312]

excelent good very good thank u

[@relike868p]

My teacher or someone of their age always complain that the checking of tables is more complicated and time-consuming than using a dictionary...

[@astrofox2409]

shouldn't sin^2 (x) be equal to 1-cos^2 (x)? because 1= sin^2 (X) +cos^2 (x)?

[@CompuBOOT]

I am not a flat-earther. Could one of the reasons flat-earthers are wrong is because they use Euclidean trigonometry instead of spherical trigonometry when determining angles and distances?