A Geometric Understanding of the Trigonometric Functions (and proof of tan𝜃 ≡ sin𝜃/cos𝜃)

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A brief look at the origins of the trigonometric functions, how to understand them geometrically, and where their names come from. Thanks for watching!
Basically this is a proof of tan𝜃 ≡ sin𝜃/cos𝜃, with some context.
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3 окт 2024

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Комментарии : 59

@Scarabola 4 года назад

I had recently learned that since Tan is defined as Sin/Cos, then the Tan of the angle also happens to be the slope of the terminal side of the angle. My mind was blown.

@BetaDude40 Год назад

By extension, -1 * cotangent would be the slope of a line orthoganal to that angle, which is really helpful when mapping reflections

@chandang5216 Год назад

I was searching for this explanation from almost 15 years Thank you so much

@lodgechant 4 месяца назад

I burst into tears when I finally understood these relationships. How breathtaking this all is. Thank you for this incredible explanation.

@iuliusconstantcornelio2018 4 года назад

The unit circle is quite a concept, really !

@niharikasrivastava3835 4 года назад

Nice teaching sir… For good English Videos , I think Vidya Guru Sonia ma'am sessions are good source. My vocab became really improved after watching her videos.

@Mochi-yr8rc 3 года назад

Why Do I learn More Math From RU-vid Then My Actual Math Teacher...?????.....

@crypticgod1134 4 года назад

Thanks for this

@perroloco83 Год назад

Thank you for this delightful explanation! 🙌👏

@bartlx Год назад

Seeing the blue tangent line drawn with a beginning and an end for a change in relation to the unit circle really helps, together with the fact the radius lands on it at 90 degrees angle.

@TechToppers 4 года назад

Was Searching for this in the whole time. Subscribing right now!😄😎

@diwitdharpatitripathi7427 3 года назад

The most important thing to learn and always remember. On the finger tips.

@fetass4089 5 месяцев назад

Why does y have a length of sine0?

@manishbhatnagar82 2 года назад

BEAUTIFUL EXPLANATION

@ach1354 4 года назад

Hello can you please go through the January 2020 mocks

@solo-moon 3 месяца назад

isent the hypotenus always the longest angle ?

@markstahl1464 5 месяцев назад

I like your video. I need a more stripped down, historical version of this for my intensive Geometry class though. Does anyone have any suggestions?

@Louis-qj4tb Год назад

God Ur incredible thank u so much!!!

@markstahl1464 5 месяцев назад

This literally makes me want to cry. The incredible intricate beauty of it. It's the three similar triangles of the Geometric Mean. Feels like a hidden message from God.

@Adityarm.08 Год назад

Very well done. Thank you.

@محمدحارثبھائی Год назад

If the length of full chord and the length of one of the arcs so formed are known then how to find the radius of the circle ?

@dvdortiz9031 Год назад

Easy.

@محمدحارثبھائی Год назад

@@dvdortiz9031 yeah but how?

@darylewalker6862 Год назад

Assuming the arc actually is circular and not from an ellipse or something else: The un-normalized sinc function is (sin x)/x. Let’s call its inverse SI(x). The chord length is 2r * sin t. The arc length is 2rt. Their ratio is sinc t. So compute SI(ratio). The quotient of that and half of the arc length is t in radians.

@gregtrinidad6655 Год назад

Circumference devided by the diameter equals radius or pi 3.1416

@shakeruso Год назад

Está un poquito mal planeado el diálogo del vídeo y eso hace que se distraiga el observador, pero la explicación en sí es muy interesante. Con un poco de planeación en el diálogo el vídeo podría durar 2 minutos menos y ser un poco más inteligible

@mosalah8166 Год назад

1:03 I don't understand why that base angle is also 90-𝜃 if clearly, it is corresponding to 𝜃.

@RalKrish Месяц назад

Take the bigger triangle into consideration

@Aleksandra-lf8ky Год назад

frajde mam kedy naduzywam wedze w/m degeneratow w branzy zwyklych zdjec

@daniels.5110 Год назад

But why is secant the one on the x axis and cosec is the one on the y axis?

@erenjager4220 2 года назад

Hi, many thanks for your videos, they are very good. Please i have few questions: On which way tan(x) can be the slope and its relationship with the derivation. And finally, how the relate the reciprocals sec(x) et cosec(x) with the diagram you've drawn: like how the reciprocal suppose to be drawn that way? many thanks

@mathonify 2 года назад

The second question I feel you can work out if you follow a similar method to the one I used for tan. Look at the right triangles created in the diagram and relate the sides. You will see the relationship between the reciprocals. The first question is interesting, I might make a seperate video on that.

@erenjager4220 2 года назад

@@mathonify yeah! I would appreciate if you make a video upon that topic! Regarding the second question; i didn’t mean how to just draw the reciprocals, but also how to interpret the 1/cos(x) (which is sec(x)) in the diagram; same for 1/sin(x); like cot(x), it’s already belonging to the tangent(theta) straight line, so what’s the difference. Many thanks for your very good work. 👏🏻

@erenjager4220 2 года назад

@@mathonify wish you make a detailed video about my second point? Appreciated

@gabriellita7236 4 года назад

Is this gcse or A level?

@mathonify 4 года назад

A level. It can also be useful for gcse but it’s not necessary

@surajksailopal 4 года назад

@@mathonify so would it be grade 9 at gcse?

@MrSeful94 4 года назад

Suraj It wouldn’t be any of the grades at all as it won’t come up

@legitstupid1683 4 года назад

@@surajksailopal I think it is included in gcse further maths. Not too sure you'd have to check.

@brahzyzz 4 года назад

@@legitstupid1683 You only need to learn some trigonometric identities in further maths (level 2), but like always learning where concepts comes from is interesting and can help.

@krishanu-d1k Год назад

Unit circle where r=1

@elinesiobrito3333 Год назад

Good job!

@Hecticweek 3 года назад

Thank You !

@AkamiChannel Год назад

This was amazing! Thank you! Just out of curiosity, where did you find this explanation?

@mathonify Год назад

I’m not sure what you mean. Which explanation?

@thabangnkopane4626 3 года назад

Proof tan (90)

@krishanu-d1k Год назад

Love from India

@drscott1 Год назад

👍🏼

@V.Z.69 3 года назад

Since you explain math, can you explain PI? PI is Diameter / Circumference, right? But how did we agree on PI as a number, a repeating decimal that supposedly goes on farther than anyone has even been able to calculate with super computers? We can't really measure a circumference nor a diameter without SOME micro-error, right? And so it's agreed upon it seems. PI seems to be an agreement of "scientism". Like how "science" agrees on gravity as a ration of Earth's mass. No proof, but its agreed upon. And if we are talking about this "explanation", where does Sine come from? Unit circle, SOHCAHTOA and all the good stuff. Let's bring up the Algorithm of Taylor's Series, or Expansion, or theorem. But that seems like an algorithm that a computer would handle, which is what a calculator does... so how did this "ancient" math happen if the algorithms are so rigorous? I mean, did a guy really write this out? It seems like a Taylor for computers, not a Taylor series to explain Sine and Cosine for "ancient man". It seems very contrived explanations of math and irrational numbers of "ancient Greek philosophers". We spend countless hours in college writing "proofs", yet the base of the math doesn't seem provable. Why would Taylor series be the definition or explanation of Sine? We have Rise / Run is a slope. Then we take a perfects slope and say that Tan(angle) is rise over run? If Rise is 3 and run is 4, then the slope is 3/4. Why do we say arcTan(3/4) is the angle then? Why is it not sufficient to say 3/4 is the slope? It seems to be only for calculations for a computer and not calculations for a man. I hope I'm making sense. I've taken math courses beyond physics and calculus and did very well. And this drives me mad. Oh just punch it into the calculator and see the Taylor series prove it... So, we didn't have computers 2 thousand years ago? I think something is very very hidden.

@mathonify 3 года назад

This is an excellent question, and something we could spend a lot of time discussing. I will only make a brief comment on irrational numbers here. I find it useful not to think of irrational numbers such as pi as an actual number, but as the lack of a ratio between two measurements. There is no scale you can use which will simultaneously measure the circumference of the circle and the diameter. You can define the diameter as "1 unit" but then the circumference you will not be able to measure with those same units to a perfect accuracy. Similarly, you could define the circumference as "1 unit" but then you cannot measure the diameter perfectly with those units. So the fact that pi is irrational is a consequence of this. The approximation of 3.14159265... is our attempt to represent the circumference and diameter on the same scale of measurement, despite the fact that it is impossible. There are many examples of these constructions that do not exist in a ratio e.g the diagonal of the square and the side of the square. It is very strange that some common geometric shapes have within them seemingly impossible dimensions, and yet we know they are there. Now whether infinity actually exists in nature is another discussion, which I won't comment on further.

@V.Z.69 3 года назад

@@mathonify That's fair. It's always driven me away, that is the explanations. "Accept this so we can just show you more 'math' "? I loved math, but the more I "learned", the more I don't believe in it's roots or historical explanations. Thank you for your time! I was agnostic before I went to college, but college proved to me that we are either liars or liars or both.

@mathonify 3 года назад

Yes and when you say the base of maths doesn't seem provable, I would agree. Take something as simple as 2+2=4. How do you prove that? I would argue that you either accept it or you don't. But it is extremely useful to accept it to be true, because when we pick up two apples, and two more apples, we have four apples. However there is nothing stopping someone from starting with 2+2=5 and building an entirely new form of arithmetic around that, as useless as that might seem.

@MidnightStorm4990 3 года назад

@@mathonify Even if you can't prove what 2 + 2 = 4, the maths that we use still works almost perfectly ;).... (I'm only in year 12 and don't even take further maths ahah)