I agree this is the best proof I've seen on you-tube. When people draw circles with unit 1 radius to recreate that same digram, you 'can't see the woods for the trees' . You drew one triangle on top of the other and all of a sudden it is clear as day.
I have a good maths teacher but his explanation was really confusing but your maths vid broke it down into tiny steps which made it so much easier to digest thank you👍
It's great to finally see a video that demonstrates several trig formulas in one diagram only. Not even to mention that if one masters that he can hence easily get a great bunch of related trig identities
I don't care about proofs, but I needed to understand visually why adding or subtracting angle ratios is done thusly and I looked at the proof somewhere else but couldn't understand it... Now I understand exctly and what's happening and I don't even need to memorize anything about compound angles.. I can draw it on the spot and derive what I need even 10 years from now!! Thank you Erik!
This is fine for 0 < A+B < 90 degrees, but what about obtuse angles when we have to use the circle definitions for trig functions. Do we need another proof or is this one enough?
First I would just remark that this proof by diagram works as long as both A and B are acute, even if 90 < A+B < 180. Next, if you have either or both A and B greater than 90, you can define A' = A - 90n and B' = B - 90m to get A+B = A'+B' + 90(n+m) where A' and B' are both acute and n+m = 1, 2, or 3. (Any multiples of 360 can be subtracted out since the sine and cosine won't change.) Now, sin(C+90) = cos(C), sin(C+180)=-sin(C), and sin(C+270)=-cos(C) and similarly for cosine. (You can verify these by coordinate geometry without having to use the addition formulas.) So, we've reduced the problem of sin(A+B) to -sin(A'+B') or ±cos(A'+B') for which the proof works. Definitely messier, yes. But I don't think you have to rely on a completely different proof.
@@erikthered109 I am glad the proof works out to the appropriate result but I think it needs to be justified why you can place the top triangle on the hypotenuse of the bottom, it's just not clearly justified in your video why the hypotenuse hit the lower triangle should equal cosine of B, if both of these triangles are on the unit circle they should share a hypotenuse of one. so if that middle line was equal to one that would be clear, I'm just confused on this, but as I said the result works out so clearly I'm missing something I just wish you had explained the diagrammatic construction but then again I'm no expert on math or making videos
Hi Anthony, thank you for your comment. I think I can understand why there might be some confusion. Only the top triangle has a hypotenuse of 1; the end of the lower triangle's hypotenuse is not on the unit circle. You may be wondering: given any right triangle on the bottom, how can I then draw a triangle with a hypotenuse of 1 on top of it, and the answer is, I can't always do that, unless the hypotenuse of the lower triangle is less than 1 unit in length. But, if the bottom triangle is too big, I can always scale down the entire diagram until the upper hypotenuse is 1; the new triangles are similar and the angles remain the same. I hope that helps clear things up.
Is this an actual proof though? Or does it only apply in this specific instance where ( kind of arbitrarily, and very conveniently to suit the ‘proof’) there is this weird shape of 2 triangles. One placed on top of the other, so one triangle’s side is the others’ hypotenuse. What if I were to devise some strange combination of shapes. Would it still work?
If we're just adding angles, the original and final x,y should be at the same distance from the origin, right? But the second triangle has a hipotenuse of 1 while the first has a hipotenuse of 1cos. Why is that?
really glad I'm not the only one questioning this because the proof ends up working out to the appropriate answer but I also don't understand why cosine of B is the hypotenuse to the lower triangle, if the lower triangle is on the unit circle it's hypotenuse should be equal to one
the reason why is the following. Imagine you want to use this to find the rotation of a vector. Lets say that the vector is the line that the guy from the video said has length cos(B). Forget about cos(B) for now. Lets say that this vector has a length of 69 for example. The vector has an angle of A degrees in respect to the X axis. What we want to find now is what the vector will be if we rotate it by B degrees until we reach an angle of A+B degrees. So, if you look at the unit circle (circle of radius 1) then we can start working from that line that has a length of 1 in this guy's video. What we are looking for when we draw the line that is perpendicular to the line with length 1 is a line that will cut with our original vector which had a lenght of 69 units. The point in which it cuts is obviously shorter than 69 units. What will be the length of the segment that we have from the origin all the way to the point where the cut happens? well, since we're working on the unit circle so that the second line has a length of 1, then the cut happens at a length of cos(B). That is why that line has that length, because it doesnt actually reach all the way to the edge of the unit circle. If we were working with a line of any other length, then the cut would happen at a different point. Imagine that the length was h, then the cut would happen at h*cos(B) which would scale the rest of the operations done in this video by h, which doesnt affect our final result but it makes working with those values more cumbersome until we reach the answer, which is why he chooses to use a length of 1. Hope this wall of text was somewhat understandable and sort of made sense.
There is a saying that Mathematics is not a hard subject. It is you who is taught by an idiot teacher. And today I found this proof in video. 3 weeks straight I couldn’t figure out this but this single video has eradicated all of the doubts and allowed me to fully underatand this. Thanks
Hi Marina, thanks for your question, it is a good one. The reason I can choose 1 as the length of the hypotenuse for the proof is because of similar triangles. If I make the hypotenuse length c, and keep the angles A and B the same, I get new similar triangles whose sides are now all multiplied in length by c. When I calculate sines and cosines, the factors of c divide out, and you get the same results for sin(A+B) and cos(A+B).
nice, I watched other proof with matrix, but I already forgot how to multiplicate with one. This one is good for me, because its geometry based proof 😅
What is the practical application of this Sin(A+B) formula and its proof. One application I can immediately think of is the possibility of finding sine of the angles like 75 degrees as an addition of two standard angles lets say 30 and 45 degree without using data or Clarke tables. Apart from that what are the real world applications of this. This trignometric identity and its proof is taught in all schools and colleges of the entire world but what is the practical application? I would greatly appreciate if you can say a few.
Hi! Great question; I and my math education colleagues often ask ourselves if there is still value in teaching trigonometric identities such as the one in this video. The fact is that there are few "real world" applications of this identity. When it was hard to calculate sines and cosines, or when you had to look up values of sine and cosine in long tables, identities such as this one may have made more sense. Modern calculators have removed this value, however. Some identities (though perhaps not this one) are certainly useful for calculus. For a longer discussion of the issue than I can provide here, see: mathmisery.com/wp/2013/05/20/memorize-or-melt-trig-identities/ But I do hope that people watching can appreciate the value of geometrical proofs.
The best and most easy and logical proof!!! Thank you 😊
i agree.
I agree this is the best proof I've seen on you-tube. When people draw circles with unit 1 radius to recreate that same digram, you 'can't see the woods for the trees' . You drew one triangle on top of the other and all of a sudden it is clear as day.
How clear and beautiful. Thank you!
I have a good maths teacher but his explanation was really confusing but your maths vid broke it down into tiny steps which made it so much easier to digest thank you👍
It's great to finally see a video that demonstrates several trig formulas in one diagram only. Not even to mention that if one masters that he can hence easily get a great bunch of related trig identities
I don't care about proofs, but I needed to understand visually why adding or subtracting angle ratios is done thusly and I looked at the proof somewhere else but couldn't understand it... Now I understand exctly and what's happening and I don't even need to memorize anything about compound angles.. I can draw it on the spot and derive what I need even 10 years from now!! Thank you Erik!
This was the best video I have seen about this proof. Thank you.
Very good Proof, thank you. I prefer this one rather than the one will a lot of fractions and assumptions, this one is much cleaner and well defined.
It's the simpliest proof I've ever seen. Thank you so much!
Hey, you were that smart kid in my Calculus 101 class in Ithaca 40 years ago. I see that you found your calling! Well done.
That's me!
this guy is good. big ups man. this is the simplest proof i've ever seen on this topic
If there are talent of teaching, this is what it is. Thank you so much for easy explanation.
Wow, that's hella intuitive. So intuitive that it was forever written into my gene!
That's the best explanation I've seen in years.
Completely brilliant. Thank you so much!
This channel deserve millions of followers
I went through 4-5 videos ...Yours is BEST
This is easiest way of this proof...you are genuine! Live long!
That's so beautifull proof! I thank you for this explanation. I haven't found this evidence in my native language. 🤗
really simple diagram-much easier to understand than other diagrams-thank you
As someone who was struggling to figure out this shit, this video is a godsend for me.
Very clear and concise - thank you!
This guy is great
Sir this is a very, and I mean very sophisticated proof that's been made so easy to understand, I thank ye
Simple and very clear. Appreciated
You made my life easier. Thank you.
Wonderfull proof wonderfully explained. Thanks
this is a very clever proof thank you
really exceptional n can be a very effective methods all thanks to ERIK. best wishes from NEPAL
Your the best maths teacher ever
Thank you, Mind-blowing explaination,very clear 😊
best and most easy proof !
Just a basic knowledge of complex numbers and Euler’s formula makes this proof almost trivial. But fascinating see the traditional real method.
This is fine for 0 < A+B < 90 degrees, but what about obtuse angles when we have to use the circle definitions for trig functions. Do we need another proof or is this one enough?
First I would just remark that this proof by diagram works as long as both A and B are acute, even if 90 < A+B < 180. Next, if you have either or both A and B greater than 90, you can define A' = A - 90n and B' = B - 90m to get A+B = A'+B' + 90(n+m) where A' and B' are both acute and n+m = 1, 2, or 3. (Any multiples of 360 can be subtracted out since the sine and cosine won't change.) Now, sin(C+90) = cos(C), sin(C+180)=-sin(C), and sin(C+270)=-cos(C) and similarly for cosine. (You can verify these by coordinate geometry without having to use the addition formulas.) So, we've reduced the problem of sin(A+B) to -sin(A'+B') or ±cos(A'+B') for which the proof works. Definitely messier, yes. But I don't think you have to rely on a completely different proof.
@@erikthered109 I am glad the proof works out to the appropriate result but I think it needs to be justified why you can place the top triangle on the hypotenuse of the bottom, it's just not clearly justified in your video why the hypotenuse hit the lower triangle should equal cosine of B, if both of these triangles are on the unit circle they should share a hypotenuse of one. so if that middle line was equal to one that would be clear, I'm just confused on this, but as I said the result works out so clearly I'm missing something I just wish you had explained the diagrammatic construction but then again I'm no expert on math or making videos
Hi Anthony, thank you for your comment. I think I can understand why there might be some confusion. Only the top triangle has a hypotenuse of 1; the end of the lower triangle's hypotenuse is not on the unit circle. You may be wondering: given any right triangle on the bottom, how can I then draw a triangle with a hypotenuse of 1 on top of it, and the answer is, I can't always do that, unless the hypotenuse of the lower triangle is less than 1 unit in length. But, if the bottom triangle is too big, I can always scale down the entire diagram until the upper hypotenuse is 1; the new triangles are similar and the angles remain the same. I hope that helps clear things up.
Thank you so much it is so clear and easy to understand
Is this an actual proof though? Or does it only apply in this specific instance where ( kind of arbitrarily, and very conveniently to suit the ‘proof’) there is this weird shape of 2 triangles. One placed on top of the other, so one triangle’s side is the others’ hypotenuse. What if I were to devise some strange combination of shapes. Would it still work?
The best video, thank you!
This is brilliant, thanks so much!
Ultimate clear explanation
wonderful proof, thank you.
omg thank you all the other videos complicate it so much!!
If we're just adding angles, the original and final x,y should be at the same distance from the origin, right? But the second triangle has a hipotenuse of 1 while the first has a hipotenuse of 1cos. Why is that?
really glad I'm not the only one questioning this because the proof ends up working out to the appropriate answer but I also don't understand why cosine of B is the hypotenuse to the lower triangle, if the lower triangle is on the unit circle it's hypotenuse should be equal to one
the reason why is the following.
Imagine you want to use this to find the rotation of a vector. Lets say that the vector is the line that the guy from the video said has length cos(B). Forget about cos(B) for now. Lets say that this vector has a length of 69 for example. The vector has an angle of A degrees in respect to the X axis. What we want to find now is what the vector will be if we rotate it by B degrees until we reach an angle of A+B degrees. So, if you look at the unit circle (circle of radius 1) then we can start working from that line that has a length of 1 in this guy's video. What we are looking for when we draw the line that is perpendicular to the line with length 1 is a line that will cut with our original vector which had a lenght of 69 units. The point in which it cuts is obviously shorter than 69 units. What will be the length of the segment that we have from the origin all the way to the point where the cut happens? well, since we're working on the unit circle so that the second line has a length of 1, then the cut happens at a length of cos(B). That is why that line has that length, because it doesnt actually reach all the way to the edge of the unit circle. If we were working with a line of any other length, then the cut would happen at a different point. Imagine that the length was h, then the cut would happen at h*cos(B) which would scale the rest of the operations done in this video by h, which doesnt affect our final result but it makes working with those values more cumbersome until we reach the answer, which is why he chooses to use a length of 1. Hope this wall of text was somewhat understandable and sort of made sense.
Perfect explanation!
I feel illuminated, thank you sir😁
super video! thumbs up.
thank you for sharing.
Ooooh thankyou King 🙏🤝🤝just Subscribed❤❤❤
Math proofs - the ones I can comprehend anyway ;-) - beautiful; and t/y for this one.
Thank you so much for this awesome video!💯👍🏻
There is a saying that Mathematics is not a hard subject. It is you who is taught by an idiot teacher. And today I found this proof in video. 3 weeks straight I couldn’t figure out this but this single video has eradicated all of the doubts and allowed me to fully underatand this. Thanks
we love you Dr. J
great job! I appreciate it.
Very helpful! Thank you
Excellent
Fantastic ,
Nice explanation ...thank you sir❤❤
Thanks for this wonderful video sir
It was really cool, THANKS
Thank you sir. So clear and beautiful..
Clear, brilliant.
Thank u so much.. its very clear..... Awesome 👍
really good and short and clear
Brilliant🍒💚
Thank you so much my friend.
Fantastic...
Very nice.Thank you
Thank you very
Good job . Thank you
Thanks a million!
God bless your soul
Very nice!
Really helpful
Thank you so much this is amazing
Of all the proofs this has to be the easiest to follow and understand.
Very nice
Love it!
Damn he's good.
That was really easy thanks, but I just have a question. What if the hypotenuse was NOT 1. How would we prove it ?
Hi Marina, thanks for your question, it is a good one. The reason I can choose 1 as the length of the hypotenuse for the proof is because of similar triangles. If I make the hypotenuse length c, and keep the angles A and B the same, I get new similar triangles whose sides are now all multiplied in length by c. When I calculate sines and cosines, the factors of c divide out, and you get the same results for sin(A+B) and cos(A+B).
excellent!
Thank you very much sir ❤
Elegant.
Wow a filty amature like myself could even understand this. great job
nice, I watched other proof with matrix, but I already forgot how to multiplicate with one. This one is good for me, because its geometry based proof 😅
Awesome.
thank you sir. ➕♾ you made my day.
Thank you this is helpful
Can we use this proof in our examinations
OMG 😢 Love you sir
Great proof. What app are you using?
I'm using an Android app called LectureNotes. The app is a bit wonky but is great for this kind of video.
What is the practical application of this Sin(A+B) formula and its proof. One application I can immediately think of is the possibility of finding sine of the angles like 75 degrees as an addition of two standard angles lets say 30 and 45 degree without using data or Clarke tables. Apart from that what are the real world applications of this. This trignometric identity and its proof is taught in all schools and colleges of the entire world but what is the practical application? I would greatly appreciate if you can say a few.
Hi! Great question; I and my math education colleagues often ask ourselves if there is still value in teaching trigonometric identities such as the one in this video. The fact is that there are few "real world" applications of this identity. When it was hard to calculate sines and cosines, or when you had to look up values of sine and cosine in long tables, identities such as this one may have made more sense. Modern calculators have removed this value, however. Some identities (though perhaps not this one) are certainly useful for calculus. For a longer discussion of the issue than I can provide here, see:
mathmisery.com/wp/2013/05/20/memorize-or-melt-trig-identities/
But I do hope that people watching can appreciate the value of geometrical proofs.
GOAT
can this diagram be used to derive cos(A-B) and sin(A-B)?
Yes: cos(A-B) = cos(A + (-B)) and sin(A-B) = sin(A + (-B)) ; use the fact that cos(-B) = cos(B) and sin(-B) = -sin(B).
Clear and concise, thank you
Thanks a lot!
So why hypotenuse of A is not 1
THANJ YOU SO MCH !!!!!!
thank you so much
Nice
It's very easy trick😊😊
Thanks a lot sir
thanks a lot