This diagram is basically a full semester of trig, and if you remember it you can derive most of the knowledge of trigonometry. For students, please remember that there is also a skill of trigonometry, which comes from repeatedly applying the knowledge. In particular, a lot of trig problems are only solvable if you recognize the different trig identities and use them to convert the form of an equation into something you can deal with. It's worthwhile to put in the practice so you can recognize these patterns.
For a full coverage you should also see a visual proof of the Pythagorean theorem, a derivation of the special angle values, and the proof of the angle sum identities (the rest of the identites follow from these and the Pythagorean ones algebraically). A higher level overview can mention their power series definitions (which generalize to nonreal inputs, like complex numbers or matrices, and are considered the formal definitions of the functions) and also Euler's formula.
Thank you, this motivated me to keep doing practice problems. I understand trig pretty decently, but I realized after even a week or two not using it I lose it all. But it really is a use it or lose it sort of math.
You are by far a better teacher than ANY of the teachers at my old high school. In just over 4 minutes, you explained in a clear concise way the fundamentals of trigonometry. Thank you!
I remember clearly back in highschool asking my teacher "what does the tan represent on the unit circle?" He said, it's just the ratio of sin and cos. Ever since then anything other than sin and cos were just equations and had no graphical meaning. +10 years later, I finally got a legitimate answer. Thanks!
How sad this must have felt, I remember when our high-school professor showed as what the tan look like on the trigonometric circle, we did not even know what is 'sec' so we could apply the pytha theorem (back then) lol
@@hmmmidkkk Isn't this a tragedy in a way? If you're meritorious you'd rather want to land up on a high paying job while teaching is considered a low profile career.. actually teaching is the most sensitive profession in social POV
I took trig in university, and understood it pretty well at the time. And I've seen the static version of this diagram, but it never really made sense to me because these functions were never taught to me this way, outside of sine and cosine. This explanation was really cool and makes a ton of sense.
Yeah that was a huge eye opener wasn't it lol. Anyone who thinks they are "bad" at trig was probably just taught by someone who had no clue how to properly teach trig. I'm going to watch this video every morning when I wake up from now on lol
@@NSNINETEEN A line is infinite, whereas a line segment is just a finite segment of the line. What you're referring to is a line segment; if you extend it infinitely, you will indeed get a secant line of the unit circle.
Imagine if every school took these functions to the simple basic level you just did in only a couple of minutes. There would be nothing scary about trig again. I wish it had all been expained so easily when I was at school. It took me to research it myself years later to understand trig. Great video.
I also just subscribed to your channel. I Was looking for proofs in video form of the sin and cosine rules for non right triangles. Do you have one, or could you make one ?
@@MathVisualProofs the sin rule ... a/sinA = b/sinB=c/sin C ... the cos rule ... a2=b2+c2-2bcCosA, or cosA=(b2+c2-a2)/2bc. I can't superscript the squares for a, b and c in the equation sorry. We were told the formula at school, but not the proofs. I always wanted to know the whys as well as the whats
@@markdonnelly1913 here’s law of cosines : Law Of Cosines II (visual proof) ua-cam.com/video/NHxJ3Z_58Lw/v-deo.html (I have another too). Don’t have law of sines yet.
As a mechanical draftsman in the 70/80s descriptive geometry using drawings technical methods was used similiar, but not knowing trig hurt my career. I had to relearn it starting with ratios...do they even teach this anymore. Oscar Had A heap Of Apples saved my butt. sin = O/H cos=A/H tan=O/A and of course pythagoreus....A2+B2=C2 Your diagram just opened my eyes - and brought it all together. Very good Sir. KISS as we would say in designing: Keep it simple stupid. Thank you.
@@MathVisualProofs When i advanced for a ME degree - the calculus 1 and 2 courses taught me how to do estimated volumes or area problems in my head. Helped in Thermo....differential was fun. TY for sharing your knowledge. A most honorable profession.
From a nerd and someone with a decent level of maths education: This is brilliant! It makes so much sense out of these concepts, all in one connected image!
I have a reasonable ability in mathematics. However to find out at the age of 61 that the co in cosine etc means complimentary is a revelation. I am somewhat surprised that that was never mentioned to me all those years ago “hay ho”. So just for that thank you very much.
LOL. I remember hitting the windshield in my 2nd Calculus class when I was suddenly confronted with the reality that I had either never learned the trig identities or had completely forgotten them. The prof was truly bad too, so instead of scrambling I decided to drop that class, review some basic math, and burned through the class the next semester. This vid might have saved the day, but this all happened in 1982.
I can't believe how poorly I was taught trig in school. I finally mastered trig on my own using the textbook "Trignometry" by Gelfand and Saul which is old but gold. UA-cam visualizations like this are the perfect supplement. Thank you!
Just remembering the agony of putting all that information into my usable knowledge !! Then remembering trying to teach that same info to my students for twenty years !! Saving and Sharing this. Blessings for the individual who put this together !
This is lit, Noone in highschool / tuitions, ever explained like this to me. I wonder why they missed such simple stufff and keep the kids, breaking their head. Many thanks for sharing!!
Your diagram really helped me with the infinity values by explaining it in simple terms like y and x never cross. I've seen the same diagram in motion but slowing it down I was able to grasp more.
I was thinking of these exact properties and interactions after seeing the static pictures and I knew some person must have animated this diagram which shows perfectly what these concepts really are. One of the most elegant math videos I've seen on UA-cam and I can't believe it's so recent.
Excellent, I love it. I was explaining trigonometry to my daughter, leveraging on my PhD in Aerospace Engineering, I looked for animation to help fix visualization of concepts, you outperformed expectetions and showed me I was not using the proper definition of tangent, in last decades.
I must admit to having to watch this slowly and think through the 'what is obvious' bits but I get it. I think it's brilliant! :-) I love the naming of 'tan' it is so obvious from the diagram. It is reminiscent of that tablet found on the beach after that massive volcanic eruption on the island of Sohcahtoa.
This is the absolutely most brilliant visual presentation of trigonometry that I've ever seen. I've seen many, many. I nominate you for the Trigonometry Nobel Prize! 😎😎😎😎
Great explanation. In school, no one explained why this formulas just like they are, but with this video I finally understood where all these formulas came from. ❤My appreciation, Gracia!
I went to high school in SoCal and all we learned from geometry is the mnemonics for the trig functions: SACAGAWEA which was based on some female Indian name. I wish I had you as teacher.
I was taught to visualize the tangent as an line tangent to the unit circle in the coordinates (1, 0). I had never seen any other representations and had a hard time trying to visualize the cossec and secant funcions 😅😅 This video is mind-blowing !! It's always great to see different ways to understand a topic
This is really nice. You should do one on the double angle theorems for sin and cos. There is a nice one that looks just like your Diophantus diagram from a few months ago.
This is how I was taught trigonometry at school. It wasn’t on the curriculum but was the best way. We had a library I would go to on the way home which had old maths books with this stuff in. Sadly I can’t find books like that anymore. You can derive the double angle formulas for sin and cos from the unit circle.
Awesome. . This video is enlightening. In the past I struggled to understand the various relationships. This VISUALZATION is so powerful.. Thanks for sharing your insights. regards / djb.
Excellent! Educational! A huge thank you! sec × sin = tan × 1 shines in my eyes pink × blue = yellow × white and this provoked me for a bit another look at secants... with memorable trigonometric "trinity" and "co-trinity" formulas :) tan = sec × sin cot = csc × cos need to say, that those are easier identifiable on "secant (ray) centric" drawing (lines x=1 and y=1 are plotted instead of tangent) which is an alternative to this, let's call it"tangent centric"
I'm very comfortable with math, having used it my entire career as a physical inorganic chemist. But always found most of the trigonometric identities to be difficult to remember (though with some algebra, I could derive them ... eventually). Honestly, I lean on Euler's relationship, exp(iθ) = cosθ + isinθ and algebra to get around the use of trigonometry quite often. I don't think I've ever seen tanθ, cotθ, secθ, and cscθ identified as line segments on the standard unit circle diagram. And why didn't I know that the "co" in cosine, cosecant, and cotangent stands for "complementary"? This a very enlightening approach!
The origin of some of the trigonometric names became clear from this diagram. tangent meaning to-touch, it is the length of the leg touching the circle. [previously I said to-kiss but that is osculate]; secant meaning to-cut, it is the length of the leg cutting the circle. co- meaning with, they are the functions which go with, or complement another. sine meaning to-curve, it is the length of the leg which follows the curve of the circle.
good thing i’m doing a trig function unit with my alg 3 class. i will refer my students to this vid during our transition from the geo ratios to the functions. thanks again!
Another way I've seen to construct tan(theta) and sec(theta) is to draw a vertical line that is tangent to the circle at the rightmost point. and to get cosec(theta) and cot(theta), draw a horizontal tangent line at the topmost point. That representation can also show all the properties show in this one. But i prefer yours cuz it's a little neater and less messy. Thank you so much. One little thing i wish you did was extend theta out of the acute range and see the trig functions in the full 2pi range, but I imagine that it might get messy, especially tan and cot. Still, great video.
Yes that’s a good idea too. Also I thought about running around the entire circle but it was a bit messy and their are some technical details to manage with supplementary angles and negative lengths :)
I would have the same suggestion, but the way it was made in the video ends up cleaner to draw. So, the suggestion using the vertical line passing through the point (1,0) would end up being kind of an interesting side note.
What you described is at wiki: "Trigonometric_functions.png". I like this one better. It is interesting that the tangent and cotangent sum to be the length of the line between the axes. The Pythagorean theorem gives the same equation as adding the last two identities: 2 + tan^2 + cot^2 = sec^2 + csc^2 That observation does not fall out of Wiki's illustration as easily.
Neat, i have always had issues with sec and csc. This diagram makes it so easy. sec(x)^2 + csc(x)^2 = (tan(x) + cot(x))^2 = tan(x)^2 + cot(x)^2 +2tan(x)cot(x) = tan(x)^2 + cot(x)^2 + 2
The fact that he explained it so calmy i felt like watching a discovery channel's documentary about an animal called trigonometry in the forest of mathematics and this guy is explaining the ferocious animal trigonometry would react when it faces its different types of prey and the prey are the different angle measures
If we assume any other radius lets say r then just multiply each identity with r to het the complete picture. Also there are two interpretations of tan, cot, sec, csc like there are two interpretations for sin amd cosine (check the diagram for two parallel vertical lines amd horizontal lines which are sin and cos. Plus when angle is 45 sin =cos , tan = cot and sec = csc. Verify from the diagram. I have a beautiful diagram on my whiteboard 😊
Great job MVP. This diagram ought to be in every geometry and trig book in America but isn't. Add: 1) automated, 2) static and 3) math experiment as a hands-on exercise to prove it to the student. Today, there is "not enough time" or "it's not in the curriculum, scope and sequence or district mandates". This with the unit circle at the key radian measurements (pi, pi/2, pi/3, etc) are the visual presentations to allow the students to understand the definitions and abstract concepts of trig. History buffs, did the definitions of trig or the diagrams come first? (20+ year retire math teacher; 16 in geometry.)
I did up a similar diagram years ago on card stock, with versine, coversine, exsecant, and excosecant included. Still haven't figure out how to intuitively include haversine, hacoversine, or the other navigational/cartographic ratios
This is a great viswualization tool. It would, hoiwever, bet easier to understand the triangle similarities in the initial setup if the angle shown were not so close to pi/4. :)
I am in Year 11 and other Pythagorean Identities I have spotted are: 1. sec²θ+csc²θ=(tanθ+cotθ)² 2. (secθ-cosθ)²=tan²θ-sin²θ 3. (cscθ-sinθ)²=cot²θ-cos²θ I am not sure if these are popular in A-Level Trigonometry since they are quite lengthy but you can probably tell I have used substitution to work out segment lengths on the unit circle diagram. Also, identities 2 and 3 are basically the same, just with complementary angles (as you mentioned in the video) since a (co)secant function and a (co)tangent function are used as the larger values on both sides of each equation, although I am not quite sure what the proof reason is for the sine and cosine functions to swap places, if you know what I mean.
Imagien you have a right triangle. The opposite side is a and the adjacent side is b and the hypotenuse is 1 as it's on thr unit circle. The sin is defined as opposite/hypotenuse and the cosine is defined as adjacent/hypotenuse. Subtitute and you get sinθ = a/1 = a, and cosθ = b/1 = b. Subtitute each side with the trig function and that's it
This just helped me figure out an entire block of undocumented JavaScript I’ve been reverse engineering that calculates rounded corners where cubic beziers meet at right angles.
one question from a high schooler who has a hard time doing math or taking concepts for granted until they feel like they understand it enough to have come up with it themselves (for whom this video has been an absolute lifesaver): how does SOH, CAH, TOA play into this? how/why does that work? i can see that in this diagram, sin(x) (gonna say that instead of theta) IS the measure of the side opposite to angle x, rather than "the opposite side over the hypotenuse". same with cos(x) (but respectively). at least TOA for tan(x) makes sense within this diagram!
When you scale the circle to have radius r, the x and y coordinates become r*cos(t) and r*sin(t) and the triangle is similar to the original. So here you can take see that sin(t) is opposite over hypotenuse because the r’s cancel in numerator and denominator. Does that make sense?
@@MathVisualProofs that does make sense! and the circle here has r = 1 and doesn't need to be scaled, so you didn't show that cancellation, is that right?
@@aktisfmthat’s right! The unit circle is used as nice circle because then the ratios work out as just sine and cosine. All circles are similar so you just need one to understand trig.
It’s good to move the point to the other quadrants to see what happens to the functions. You can also prove double angle identities using the unit circle.
fantantisc video. that diagram is very convenient. It is compound of 7 right triangles, all similars. and you can apply a scalar factor to transform the original triangle in every six others. Of course Pythagorean theorem gives interesting identities too. Other way to plot the length of tan theta, sec theta is intersecting line y = x · tan (theta) with line x = 1 at point (1, tan theta) what is equivalent to applying to original triangle a factor of sec theta.
hi could you please clarify when tangent is drawn with circle perpendicular to radius with statement that "because these two picture trinagles are similar" ?(time:0.55) . The one angle is similar of 90 in both and common line (sine theta one), what else? how come they similar. what concludes it to be similar.
@@MathVisualProofs How can we say that "and they have theta in common"? I think I got this part, there might be gap in explanation at 0:58, when ratios of their sides are equal then it should not be d/1 = sinx/cosx, but rather d/sinx = 1/cosx which then simplified to d/1=sinx/cosx. btw, it will be more clear if triangles vertex are named, so one is inside of circle and other is the bigger one (includes the smaller inner one with tangent). the query is for second cot definition, where angles are not even same theta, but pi/2-theta. so not similar one, then how can triangles are similar for side to be compared.
Some Officers Have Coaches And Horses To Order Around Sin = Opposite/Hypotenuse Cos = Adjacent/Hypotenuse Tan = Opposite/Adjacent That's how I was told to remember it at school. I'm 59.
Weirdly enough, I don't know if I should acknowledge this but this diagram was explained in the byjus class 10 science videos back in 2017. Those days they were really good.
i’ve never seen this diagram…even a static version is quite informative, but the animation knocks it out the park…thanks!
👍😀
The static version would have been clearer still if the angle was not so close to 45°, so that the sine and cosine are virtually the same value.
@@bart2019 yes. Hard to make space for everything. But the angle is not 45 :)
I've seen it before in a math encyclopedia (but never an actual textbook) but seeing it animated like this is far better.
@@Mark73 😀👍
This diagram is basically a full semester of trig, and if you remember it you can derive most of the knowledge of trigonometry.
For students, please remember that there is also a skill of trigonometry, which comes from repeatedly applying the knowledge. In particular, a lot of trig problems are only solvable if you recognize the different trig identities and use them to convert the form of an equation into something you can deal with. It's worthwhile to put in the practice so you can recognize these patterns.
For a full coverage you should also see a visual proof of the Pythagorean theorem, a derivation of the special angle values, and the proof of the angle sum identities (the rest of the identites follow from these and the Pythagorean ones algebraically). A higher level overview can mention their power series definitions (which generalize to nonreal inputs, like complex numbers or matrices, and are considered the formal definitions of the functions) and also Euler's formula.
Law of sines and cosines is also good to prove, although it's more a fact about triangles than just the trigonometric functions themselves.
Trig identities turn math into magic at higher levels. I am no shin lim, lets just say.
So many people are scared of this , what a waste
Thank you, this motivated me to keep doing practice problems. I understand trig pretty decently, but I realized after even a week or two not using it I lose it all. But it really is a use it or lose it sort of math.
Suddenly the complimentary functions and the identities make so much sense. Brilliant way of showing all the trig functions.
👍😀
💯💯💯💯
You are by far a better teacher than ANY of the teachers at my old high school. In just over 4 minutes, you explained in a clear concise way the fundamentals of trigonometry. Thank you!
Thanks for the kind words. Glad this helped.
I remember clearly back in highschool asking my teacher "what does the tan represent on the unit circle?" He said, it's just the ratio of sin and cos. Ever since then anything other than sin and cos were just equations and had no graphical meaning.
+10 years later, I finally got a legitimate answer. Thanks!
Glad to help!
How sad this must have felt, I remember when our high-school professor showed as what the tan look like on the trigonometric circle, we did not even know what is 'sec' so we could apply the pytha theorem (back then) lol
Holy crap. It has only taken me 60 years to stumble across this explanation.
Better late than never. :)
I learnt trigonometry since highschool.. WHY DIDN'T THEY TEACH THIS EVER!? It's so easy and sensible in this way.. why!!??
Because they themselves don't understand the concept that's why they're teaching and not in a higher paying job
@@hmmmidkkk Isn't this a tragedy in a way? If you're meritorious you'd rather want to land up on a high paying job while teaching is considered a low profile career.. actually teaching is the most sensitive profession in social POV
@@shreeniwaz kulukulu is the reason
@@hmmmidkkk I would be wary of painting all teachers with the same brush. Ludwig Wittgenstein was a teacher, and Bertrand Russell was a lecturer.
they were honing their skills to teach gender studies and critical race theory
I took trig in university, and understood it pretty well at the time. And I've seen the static version of this diagram, but it never really made sense to me because these functions were never taught to me this way, outside of sine and cosine.
This explanation was really cool and makes a ton of sense.
Glad you liked it!
Jesus, no one has ever explained to me why that darn function is called tangent. Thank you.
Yeah that was a huge eye opener wasn't it lol. Anyone who thinks they are "bad" at trig was probably just taught by someone who had no clue how to properly teach trig. I'm going to watch this video every morning when I wake up from now on lol
But it got mw in another confusion i,e why is that line at the base called secent as it does not intersect the circle at two points
@@NSNINETEEN A line is infinite, whereas a line segment is just a finite segment of the line. What you're referring to is a line segment; if you extend it infinitely, you will indeed get a secant line of the unit circle.
Imagine if every school took these functions to the simple basic level you just did in only a couple of minutes. There would be nothing scary about trig again. I wish it had all been expained so easily when I was at school. It took me to research it myself years later to understand trig. Great video.
I also just subscribed to your channel. I Was looking for proofs in video form of the sin and cosine rules for non right triangles. Do you have one, or could you make one ?
Can you specify which rules you mean? I do have a trig playlist, but I can take requests and see what I can do.
@@MathVisualProofs the sin rule ... a/sinA = b/sinB=c/sin C ... the cos rule ... a2=b2+c2-2bcCosA, or cosA=(b2+c2-a2)/2bc. I can't superscript the squares for a, b and c in the equation sorry. We were told the formula at school, but not the proofs. I always wanted to know the whys as well as the whats
@@markdonnelly1913 here’s law of cosines : Law Of Cosines II (visual proof)
ua-cam.com/video/NHxJ3Z_58Lw/v-deo.html (I have another too). Don’t have law of sines yet.
As a mechanical draftsman in the 70/80s descriptive geometry using drawings technical methods was used similiar, but not knowing trig hurt my career. I had to relearn it starting with ratios...do they even teach this anymore.
Oscar Had A heap Of Apples saved my butt. sin = O/H cos=A/H tan=O/A and of course pythagoreus....A2+B2=C2
Your diagram just opened my eyes - and brought it all together. Very good Sir. KISS as we would say in designing: Keep it simple stupid. Thank you.
👍
Oscar had a heap of apples . . . Yes, but if you're trying to teach adolescent boys, I suggest: Sam's Old Horse Could Always Hump The Old Ass
@@MathVisualProofs When i advanced for a ME degree - the calculus 1 and 2 courses taught me how to do estimated volumes or area problems in my head. Helped in Thermo....differential was fun. TY for sharing your knowledge. A most honorable profession.
@ 😀
From a nerd and someone with a decent level of maths education: This is brilliant! It makes so much sense out of these concepts, all in one connected image!
👍😀
heh
I have a reasonable ability in mathematics. However to find out at the age of 61 that the co in cosine etc means complimentary is a revelation. I am somewhat surprised that that was never mentioned to me all those years ago “hay ho”. So just for that thank you very much.
😀👍
my complements to you sir
@@jasonrubik why do i find this so funny 😂😂
Incredibly helpful. It bridged geometry into trig for me. Trig makes so much sense now.
Glad it helped!
LOL. I remember hitting the windshield in my 2nd Calculus class when I was suddenly confronted with the reality that I had either never learned the trig identities or had completely forgotten them. The prof was truly bad too, so instead of scrambling I decided to drop that class, review some basic math, and burned through the class the next semester. This vid might have saved the day, but this all happened in 1982.
I can't believe how poorly I was taught trig in school. I finally mastered trig on my own using the textbook "Trignometry" by Gelfand and Saul which is old but gold. UA-cam visualizations like this are the perfect supplement. Thank you!
Just remembering the agony of putting all that information into my usable knowledge !! Then remembering trying to teach that same info to my students for twenty years !! Saving and Sharing this. Blessings for the individual who put this together !
Thanks!
thank you for gifting me this after I just started precalc
This is lit, Noone in highschool / tuitions, ever explained like this to me. I wonder why they missed such simple stufff and keep the kids, breaking their head. Many thanks for sharing!!
Glad it was helpful!
Self learning is one of the best things you can ever acquire and inspire to do..
So clear, concise, and compact as well, this video is a gem. Many thanks for providing it!
Thanks for watching!
This is one of the BEST math videos I've ever seen!! THANK YOU!! THANK YOU!! THANK YOU!!
Glad it was helpful!
The diagram is really intuitive
Your diagram really helped me with the infinity values by explaining it in simple terms like y and x never cross.
I've seen the same diagram in motion but slowing it down I was able to grasp more.
Excellent!
I was thinking of these exact properties and interactions after seeing the static pictures and I knew some person must have animated this diagram which shows perfectly what these concepts really are.
One of the most elegant math videos I've seen on UA-cam and I can't believe it's so recent.
Thanks!
Excellent, I love it. I was explaining trigonometry to my daughter, leveraging on my PhD in Aerospace Engineering, I looked for animation to help fix visualization of concepts, you outperformed expectetions and showed me I was not using the proper definition of tangent, in last decades.
This isn’t necessarily the proper one. It’s just one way to visualize tangent (still sine over cosine no matter what). Glad this helped!
I must admit to having to watch this slowly and think through the 'what is obvious' bits but I get it. I think it's brilliant! :-) I love the naming of 'tan' it is so obvious from the diagram. It is reminiscent of that tablet found on the beach after that massive volcanic eruption on the island of Sohcahtoa.
Glad you liked it.
This is the absolutely most brilliant visual presentation of trigonometry that I've ever seen. I've seen many, many. I nominate you for the Trigonometry Nobel Prize! 😎😎😎😎
Mind Blown! First time grasping why these things are the case instead of pure memorization
this is by far one of the most important thing that I've learnt on you tube
😀
Great explanation. In school, no one explained why this formulas just like they are, but with this video I finally understood where all these formulas came from. ❤My appreciation, Gracia!
nice job！
Thanks!
Wow, after watching this video I believe I can never miss out keeping the trig identities in memory!
Thanks!
Wow! Thank you too!
Basically a whole chapter of trig is summed up beautifully by this diagram and its labels
Glad it helps supplement the text!
I went to high school in SoCal and all we learned from geometry is the mnemonics for the trig functions: SACAGAWEA which was based on some female Indian name. I wish I had you as teacher.
Do you mean SOH CAH TOA ?
THAT'S what all those buttons I never use on my calculator are...thank you!
I was taught to visualize the tangent as an line tangent to the unit circle in the coordinates (1, 0). I had never seen any other representations and had a hard time trying to visualize the cossec and secant funcions 😅😅 This video is mind-blowing !! It's always great to see different ways to understand a topic
This is really nice. You should do one on the double angle theorems for sin and cos. There is a nice one that looks just like your Diophantus diagram from a few months ago.
Yes! I have a couple planned/in the works. Slowly but surely I'll get around to them I think.
What a neat and to-the-point representation! Thank you so much!
Thanks for checking it out!
Thanks a lot professor I follow you from Algeria
Glad to have you here!
This is a very creative explanation, and the animation brings it to life. Well done!
Thank you very much!
This is how I was taught trigonometry at school. It wasn’t on the curriculum but was the best way. We had a library I would go to on the way home which had old maths books with this stuff in. Sadly I can’t find books like that anymore. You can derive the double angle formulas for sin and cos from the unit circle.
Woooow I have an impermeable brain , but this video finally made it porous! Thanks 🙏
Wow! Mind Blown🔥. Why doesn't this diagram still not available in 🇮🇳 text books.
Is this the best video ever on maths??
Thanks! 😎
It's strange because mathematicians used this schematic to prove the Pythagoras theorem and Morley's Trisector theorem historically.
Awesome. . This video is enlightening. In the past I struggled to understand the various relationships. This VISUALZATION is so powerful.. Thanks for sharing your insights. regards / djb.
Glad it was helpful! Thanks for checking it out.
I sat in a 2 hour class not understanding a single thing, just for this visual to teach me in less than 5 minutes. thank you!
😀👍
Gracias por la Gran explicación. La proyección de la tangente fuera de la circunferencia, me dió otra perspectiva de las funciones.
Excellent! Educational! A huge thank you!
sec × sin = tan × 1
shines in my eyes
pink × blue = yellow × white
and this provoked me for a bit another look at secants...
with memorable trigonometric "trinity" and "co-trinity" formulas :)
tan = sec × sin
cot = csc × cos
need to say, that those are easier identifiable on "secant (ray) centric" drawing (lines x=1 and y=1 are plotted instead of tangent) which is an alternative to this, let's call it"tangent centric"
👍😀
Best explanation ever
Best trig description ever made
Dude, you're doing the "good Lord's work"!!! Wow.
This is one of the most important vids on UA-cam.
Appreciate this comment!
I was unable to visualize all of the non hyperbolic tangent functions before this video. Great stuff!
Glad this helped!
1:20 can someone explain :/
OKAY WOW
SO MUCH INFO COMPRESSED INTO 4 MINS
Longer version on my channel isn’t so compressed :)
Thank you and greetings from Brazil!
👍
Dude. Very well done. This static diagram as a poster should be standard trig classroom accessories!
Agree. I wonder if it exists…
I sweare Your animation thinking is next level
Thanks!
You’re doin the Lord’s work brother! Keep it up!
Thanks!
More proof that most of us are not horrible at math , but rather most math teachers are horrible at teaching.
I'm very comfortable with math, having used it my entire career as a physical inorganic chemist. But always found most of the trigonometric identities to be difficult to remember (though with some algebra, I could derive them ... eventually). Honestly, I lean on Euler's relationship, exp(iθ) = cosθ + isinθ and algebra to get around the use of trigonometry quite often.
I don't think I've ever seen tanθ, cotθ, secθ, and cscθ identified as line segments on the standard unit circle diagram. And why didn't I know that the "co" in cosine, cosecant, and cotangent stands for "complementary"? This a very enlightening approach!
The origin of some of the trigonometric names became clear from this diagram.
tangent meaning to-touch, it is the length of the leg touching the circle. [previously I said to-kiss but that is osculate];
secant meaning to-cut, it is the length of the leg cutting the circle.
co- meaning with, they are the functions which go with, or complement another.
sine meaning to-curve, it is the length of the leg which follows the curve of the circle.
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thank you so much for this gem
Glad you liked it!
I recently learned about the unit circle in school. But only about sine and cosine, not all six of them. Thanks for making this video.
Glad it was helpful!
In India we indeed are being taught trigonometry in this circular fashion. Nice visualisation. Thanks.
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Thanks for sharing this information. It really helped me to understand trigonometry better. Hope you can create more videos like this
This made me visualise trigonometry better than anything else ever could
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good thing i’m doing a trig function unit with my alg 3 class. i will refer my students to this vid during our transition from the geo ratios to the functions. thanks again!
Glad it can help!
Just BRILLIANT knowledge and imagination as well ! Many thanks for sharing , okay ?
Thank you!
For cot(θ), you can also use the idea of alternate interior angles being congruent to help you along the way!
Nice!
@@MathVisualProofs Thanks! I do have a BS degree in math, btw!
Subscribed...new way of looking into trig
Thanks! Glad you liked it :)
Finally! "Co-" makes sense! 😁
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This is the best way to learn trigonometry. period
Another way I've seen to construct tan(theta) and sec(theta) is to draw a vertical line that is tangent to the circle at the rightmost point. and to get cosec(theta) and cot(theta), draw a horizontal tangent line at the topmost point. That representation can also show all the properties show in this one. But i prefer yours cuz it's a little neater and less messy. Thank you so much.
One little thing i wish you did was extend theta out of the acute range and see the trig functions in the full 2pi range, but I imagine that it might get messy, especially tan and cot. Still, great video.
Yes that’s a good idea too. Also I thought about running around the entire circle but it was a bit messy and their are some technical details to manage with supplementary angles and negative lengths :)
I would have the same suggestion, but the way it was made in the video ends up cleaner to draw. So, the suggestion using the vertical line passing through the point (1,0) would end up being kind of an interesting side note.
What you described is at wiki: "Trigonometric_functions.png". I like this one better. It is interesting that the tangent and cotangent sum to be the length of the line between the axes.
The Pythagorean theorem gives the same equation as adding the last two identities: 2 + tan^2 + cot^2 = sec^2 + csc^2
That observation does not fall out of Wiki's illustration as easily.
MVP: gives a detailed explanation about trigonometry and equations
Me: "Mmm, yes. Circle is made from triangles, which make more triangles"
Neat, i have always had issues with sec and csc. This diagram makes it so easy.
sec(x)^2 + csc(x)^2 = (tan(x) + cot(x))^2
= tan(x)^2 + cot(x)^2 +2tan(x)cot(x)
= tan(x)^2 + cot(x)^2 + 2
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The fact that he explained it so calmy i felt like watching a discovery channel's documentary about an animal called trigonometry in the forest of mathematics and this guy is explaining the ferocious animal trigonometry would react when it faces its different types of prey and the prey are the different angle measures
If we assume any other radius lets say r then just multiply each identity with r to het the complete picture. Also there are two interpretations of tan, cot, sec, csc like there are two interpretations for sin amd cosine (check the diagram for two parallel vertical lines amd horizontal lines which are sin and cos. Plus when angle is 45 sin =cos , tan = cot and sec = csc. Verify from the diagram. I have a beautiful diagram on my whiteboard 😊
Thanks a lot 🙏
Surely gonna help a lot in mechanics.
Great job MVP. This diagram ought to be in every geometry and trig book in America but isn't. Add: 1) automated, 2) static and 3) math experiment as a hands-on exercise to prove it to the student. Today, there is "not enough time" or "it's not in the curriculum, scope and sequence or district mandates". This with the unit circle at the key radian measurements (pi, pi/2, pi/3, etc) are the visual presentations to allow the students to understand the definitions and abstract concepts of trig. History buffs, did the definitions of trig or the diagrams come first? (20+ year retire math teacher; 16 in geometry.)
Thanks!
I did up a similar diagram years ago on card stock, with versine, coversine, exsecant, and excosecant included.
Still haven't figure out how to intuitively include haversine, hacoversine, or the other navigational/cartographic ratios
This is a great viswualization tool. It would, hoiwever, bet easier to understand the triangle similarities in the initial setup if the angle shown were not so close to pi/4. :)
Yes. I tried various angles and it was hard to read either sin or cos depending on the angle. So I went close to pi/4 (though not pi/4). Thanks!
I agree, now one can confuse the similarities of triangles for being identical.
I am in Year 11 and other Pythagorean Identities I have spotted are:
1. sec²θ+csc²θ=(tanθ+cotθ)²
2. (secθ-cosθ)²=tan²θ-sin²θ
3. (cscθ-sinθ)²=cot²θ-cos²θ
I am not sure if these are popular in A-Level Trigonometry since they are quite lengthy but you can probably tell I have used substitution to work out segment lengths on the unit circle diagram. Also, identities 2 and 3 are basically the same, just with complementary angles (as you mentioned in the video) since a (co)secant function and a (co)tangent function are used as the larger values on both sides of each equation, although I am not quite sure what the proof reason is for the sine and cosine functions to swap places, if you know what I mean.
AMAZING CONCEPT VISUALISATION THROUGHT THE. STATIC DIAGRAM AND FUNCTION👍
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Can someone explain how he took sintheta/costheta on 00:46
It is a tangent so it will be perpendicular. And in the big triangle has angle thetha. So the big triangle is similar to the small triangle.
Imagien you have a right triangle. The opposite side is a and the adjacent side is b and the hypotenuse is 1 as it's on thr unit circle. The sin is defined as opposite/hypotenuse and the cosine is defined as adjacent/hypotenuse. Subtitute and you get sinθ = a/1 = a, and cosθ = b/1 = b. Subtitute each side with the trig function and that's it
This just helped me figure out an entire block of undocumented JavaScript I’ve been reverse engineering that calculates rounded corners where cubic beziers meet at right angles.
one question from a high schooler who has a hard time doing math or taking concepts for granted until they feel like they understand it enough to have come up with it themselves (for whom this video has been an absolute lifesaver):
how does SOH, CAH, TOA play into this? how/why does that work? i can see that in this diagram, sin(x) (gonna say that instead of theta) IS the measure of the side opposite to angle x, rather than "the opposite side over the hypotenuse". same with cos(x) (but respectively). at least TOA for tan(x) makes sense within this diagram!
When you scale the circle to have radius r, the x and y coordinates become r*cos(t) and r*sin(t) and the triangle is similar to the original. So here you can take see that sin(t) is opposite over hypotenuse because the r’s cancel in numerator and denominator. Does that make sense?
@@MathVisualProofs that does make sense! and the circle here has r = 1 and doesn't need to be scaled, so you didn't show that cancellation, is that right?
@@aktisfmthat’s right! The unit circle is used as nice circle because then the ratios work out as just sine and cosine. All circles are similar so you just need one to understand trig.
It’s good to move the point to the other quadrants to see what happens to the functions. You can also prove double angle identities using the unit circle.
fantantisc video. that diagram is very convenient. It is compound of 7 right triangles, all similars. and you can apply a scalar factor to transform the original triangle in every six others. Of course Pythagorean theorem gives interesting identities too.
Other way to plot the length of tan theta, sec theta is intersecting line y = x · tan (theta) with line x = 1 at point (1, tan theta) what is equivalent to applying to original triangle a factor of sec theta.
Yes! That ways is nice too. Better in some respects :)
Salute to your efforts good sir.
Thanks!
hi could you please clarify when tangent is drawn with circle perpendicular to radius with statement that "because these two picture trinagles are similar" ?(time:0.55) . The one angle is similar of 90 in both and common line (sine theta one), what else? how come they similar. what concludes it to be similar.
Both are right triangles and they have theta in common. So that means all three angles match.
@@MathVisualProofs How can we say that "and they have theta in common"? I think I got this part, there might be gap in explanation at 0:58, when ratios of their sides are equal then it should not be d/1 = sinx/cosx, but rather d/sinx = 1/cosx which then simplified to d/1=sinx/cosx. btw, it will be more clear if triangles vertex are named, so one is inside of circle and other is the bigger one (includes the smaller inner one with tangent). the query is for second cot definition, where angles are not even same theta, but pi/2-theta. so not similar one, then how can triangles are similar for side to be compared.
The best trig diagram. Thanks ❤
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I finally understand Trigonometry!🤗
I’m an Electrical Engineering Technician.
I dropped out of college because of Calculus.
Now l understand.
Glad this helped!
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Sin = Opposite/Hypotenuse
Cos = Adjacent/Hypotenuse
Tan = Opposite/Adjacent
That's how I was told to remember it at school. I'm 59.
Another one might be:
csc^2(theta) + sec^2(theta) = {cot(theta) + tan(theta)}^2
sin2α = 2sinα * cosα
cos2α = cos^2(α) - sinα^2(α)
tan2α = 2tanα / 1-tan^2(α)
A nonstandard one for sure.
Not sure I see these directly here.
also 1 + tan^2 = sec^2 and 1 + cot^2 = csc^2 then 2.tan.cot = 2 so tan.cot = 1 therefore 0.infinity = 1 )
Now I finally know why they are called the Tangent and Cotangents , They ARE Literally what they are called
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Thank you so much for this masterpiece
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Weirdly enough, I don't know if I should acknowledge this but this diagram was explained in the byjus class 10 science videos back in 2017. Those days they were really good.
This diagram is awesome.
:)
Really neat visual representation of all the fundamental angle 📐 concepts. Can I ask what you used to create the visuals
I use manimgl for these animations.