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.
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.
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.
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.
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
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.
@@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. 👏🏻
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.
@@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.
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
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.
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.
@@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.
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.
@@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)
I burst into tears when I finally understood these relationships. How breathtaking this all is. Thank you for this incredible explanation.
I was searching for this explanation from almost 15 years
Thank you so much
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.
By extension, -1 * cotangent would be the slope of a line orthoganal to that angle, which is really helpful when mapping reflections
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.
The unit circle is quite a concept, really !
Thank you for this delightful explanation! 🙌👏
Was Searching for this in the whole time. Subscribing right now!😄😎
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.
So what is the green line? Does it have a function?
Or is it just secø - cosø?
Yes sec - cos. Or versin + exsec… but sec - cos is surely more common
Thankyou really helpful
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 ?
Easy.
@@dvdortiz9031 yeah but how?
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.
Circumference devided by the diameter equals radius or pi 3.1416
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?
Hello can you please go through the January 2020 mocks
BEAUTIFUL EXPLANATION
isent the hypotenus always the longest angle ?
But why is secant the one on the x axis and cosec is the one on the y axis?
Why does y have a length of sine0?
The most important thing to learn and always remember. On the finger tips.
Thanks for this
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
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.
@@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. 👏🏻
@@mathonify wish you make a detailed video about my second point? Appreciated
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.
Very well done. Thank you.
1:03 I don't understand why that base angle is also 90-𝜃 if clearly, it is corresponding to 𝜃.
Take the bigger triangle into consideration
Why Do I learn More Math From TH-cam Then My Actual Math Teacher...?????.....
Is this gcse or A level?
A level. It can also be useful for gcse but it’s not necessary
@@mathonify so would it be grade 9 at gcse?
Suraj It wouldn’t be any of the grades at all as it won’t come up
@@surajksailopal I think it is included in gcse further maths. Not too sure you'd have to check.
@@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.
This was amazing! Thank you! Just out of curiosity, where did you find this explanation?
I’m not sure what you mean. Which explanation?
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
Good job!
God Ur incredible thank u so much!!!
Thank You !
Unit circle where r=1
👍🏼
Proof tan (90)
Love from India
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.
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.
@@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.
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.
@@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)
💀
frajde mam kedy naduzywam wedze w/m degeneratow w branzy zwyklych zdjec