Angle sum identities for sine and cosine
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- Опубліковано 26 гру 2017
- use complex numbers: • Angle Sum formula, pro...
proof of angle sum formulas for sine and cosine,
sin(a+b)=sin(a)cos(b)+cos(a)sin(b),
cos(a+b)=cos(a)cos(b)-sin(a)sin(b),
triangle proof,
angle addition identity of sine,
angle addition identity of cosine,
blackpenredpen,
math for fun,
I used to hatemath and it was the thing I hated the most.
Now I watch videos of math everyday, when I have time I take a paper and try to do the same things and follow up. And I'm always waiting for a new video to come up.
Thank you so much for making me love math
Adil D.
What a powerful comment!! I am very very happy to hear this! Thank you!!!
blackpenredpen I seriously don't know how to thank you. And I'm pretty sure a majority of people feel the same way as I do.
A BIG Thanks.
same
I am a high school math teacher with a masters degree in math and the visual brilliance and explanation of this proof is top notch. Definitely can be used to help my higher level math students. In fact, I can use it just to help regular students REFLECT on the brilliance and challenge of coming up with a good proof. Nice JOB!
Same here bro, internet is much more useful than school
This video does not deserve any single downvote. This is one of the best videos which definitely benefits secondary school students quite a lot in understanding the trigo world.
downvote? are we living in aops?
@@aasthasharma3820 You mean reddit?
FY. It plain out sux - dude don't know what the heII he is talking about...
What a fantastic proof, youve quickly become my favorite channel
Drake Alexander Golden thank you!!!!
103 Book?
The text book we use offers a 'proof' that's tough to follow, and even tougher to reproduce. Your method is great, and with practice, I've gotten it down to just over 3 minutes. The fact that it offers both the Sine and Cosine addition equations in one proof just adds to its beauty.
Keep these coming!
0:03 - When I simplify an expression over the length of multiple pages just to arrive at what I started from.
one of the most beautiful proofs, it seems perfectly designed to tease out that double angle formula in the most efficient way! But, is there no channel out there which explains how to design proofs like this? This kind of feels like what experimentalists go through to set up the perfect experiment.
He does a really great job of explaining stuff all the way. You can tell he's having a really great time doing it also. Using this for my students tomorrow!
12:20 what happened did your true form almost reveal itself
mrBorkD oh good it wasn't just me! 😂😂😂
@@aervanath me too
this guy truly loves math
your channel is cool too, just checked it, keep up!
means a lot, thanks!
You could also use Euler formula : exp(iA) = cosA + isinA and exp(i(A+B)) = exp(iA)*exp(iB) ---->
cos(A+B) + isin(A+B) = (cosA + isinA)(cosB + isinB) ---> expand and equalize real part and imaginary part --->
cos(A+B) =... and sin(A+B)=...
Actually all trig identities can be derived from Euler's formula.
Thats acute proof
Ah ha!
This is indeed a very cute proof
Rodrigo Ângelo yay!!!! I am glad that you like it!
It's been over 30 years since my trig class. I watched this to teach my kids during the pandemic (no school). Thanks for the great video.
I am with you VietVF. I am definitely using this video as an intro to teaching trig identities.
Excellent video. I appreciate it when teacher actually go through the effort to explain why these formulas are correct and how to do the actually proofs so that the students can actually begin to grasp the concepts behind math rather just simply memorizing formulas.
I love you! Please make videos like this again. You're so advanced in your new videos that I also like.
Thank you so very much for this video. I didn't like math at high school but now, 10 years later, I am rediscovering the beauty of it!
That was absolutely fantastic. I did not get lost in any part during that. Clear as Crystal! Hope to see loads more content like this!
I almost failed math in highschool and calculus held me back a year in university. But for the last six months I've been re-teaching myself through khan academy & UA-cam. I now understand things deeper than how they were taught to me. It's a shame that so many who like me fell through the gaping cracks of a system that doesn't care may not be able to muster up the courage or find the time and energy to try again. But if you're someone who's wanted to see if they could try again on their own, I believe that without much time you'll do much better than you ever have and with a lot less stress and anxiety and if you don't know where to start Khan Academy's Algebra 1 & 2 courses are very thorough, building from the absolute foundation and very understandable. Good luck.
This is an excellent presentation of a very neat proof, which you had essentially already shown in a recent video on calculating the trig ratios of 75°. Just one little point: the value of the angle alpha+beta follows immediately from alternating angles.
Michael Rothwell
Thank you Michael and omg yes I didn't see it when I was recording the video :)
You're welcome!
By the way, I've done quite a few diagrams today and concluded that your geometric proof (given for the case acute+acute=acute) can be adapted to work in nearly all other cases, even for negative angles, as described below.
Here is the general construction:
1. Measure an angle α anticlockwise from the positive x-axis, and draw the line l through O in this (and the opposite) direction.
2. Measure an angle α+β anticlockwise from the positive x-axis, and mark the point B at 1 unit from O in this direction.
3. Drop a perpendicular from B onto the line l, meeting it at A (this may be in the direction α from O or the opposite direction)
4. We now have a ΔAOB. Draw the smallest possible rectangle with sides parallel to the axes enclosing this triangle: the left side will be x=(smallest x coordinate of A, O, B), the right side will be x=(largest x coordinate of A, O, B) and similarly for the top and bottom sides (replacing x by y in the these descriptions).
With this construction complete, ΔAOB will be complemented three right-angled triangles inside the rectangle, having as their hypotenuses the three sides OA, AB, and BO, and horizontal and vertical segments as their legs:
- The side OA will make an acute angle α' with the horizontal side of its right-angled triangle, where α' is the "auxiliary angle" for α, that is the acute angle between the line l and the x-axis (algebraically, α' is the unique acute angle such that α can be written as α=n180°+α' or α=n180°-α').
- The side AB (perpendicular to OA by construction) will make the same acute angle α' with the vertical side of its right-angled triangle.
- The side BO will make the acute angle (α+β)' with the horizontal side of its right-angled triangle, where (α+β)' is the auxiliary angle for α+β.
In the right-angled triangle ΔAOB itself, the angle
Starting with the unit diagonal OB = 1, we get AB=sinβ' which equals either sinβ or -sinβ (according to the quadrant in which β lies; the sign can be readily checked from the unit circle definition of the trig functions) and OA=cosβ, which equals either cosβ or -cosβ (note that one should label the sides in terms of a signed trig function of β, not in terms of β').
Next, and using the above information about the angles of the three complementary right-angled triangles, we can calculate the six legs of these triangles in terms of trig functions of α & β, which will be just as before, except for possible minus signs.
Finally, we equate expressions for the two vertical sides for the angle sum formula for sin(α+β) and those for the two horizontal sides for the angle sum formula for cos(α+β), at which point, and despite any minus signs that appeared along the way, should be the same formulae as in the acute+acute=acute case.
You know,you are the reason why I am enjoying maths,keep your good work up man! Awesome!
This is the best proof I've found on UA-cam for the sum and difference identities.
This is not only a neat proof, it also helps me to remind that formula pretty well :D
This was the best explanation ever! I finally understood this after 16 years! THANK YOU!
I've been searching for an explanation for a very similar proof I copied down and figured I could just find out why some of the sides had those multiplied functions of sine and cosine, but I got a little too impatient and decided I NEEDED to know why. Thank you so much for this clear explanation!!!!
Excellant Lesson!!!!!! I appreciate it. It helps to see where the 2 identities come from.
Very cool proof! I’ve only ever been able to make sense of the formula using complex exponentials, but this is far more visually intuitive.
What a Fantabulous🎍 proof....🌋now your channel is my favorite channel.....🎊🎊keep moving🎉🎉🎋forward...🎋
A very great job! Well-done. Concisely done
We all know this proof very well but some of them in this earth can deliver in a very easy manner and you are one of them among all,
Appreciate your work🙏🙏
A beautiful Christmas present !
I sould recognize that i use this formula whithout surching the proof. Thank you very much !!!!
Beautifully done! I enjoy all your videos.
I've seen a lot of proofs of these results, but this is by far the best. I'll use tomorrow to show my class, and you'll get full credit for it! I'll let you know how it goes. Thanks so much.
: )Thank you~!
@@blackpenredpen so.. did he let u know?
Yeah, did he?
On the unit circle I know what sin and cosine give, but I’ve never got why sine over cosine would give the length of the tangent. My teacher never explained trig functions at all, when I went through GCSE they were known to me as “those awkward functions”
What a great proof ... finally a way for me to remember it quickly the right way around
Your videos are sooooo great!! Thank you!
Absolutely beautiful.
Thank you! One video solves trigonometry problem that I had going through for years in my head...
This is a very nice proof. Love it. :)
Beautifully done!!!
This is so brilliant! I love it!!
Very creative! Amazing proof!
Explore this construction through this codepen: codepen.io/anon/pen/bLXbmV (I made it in like 5 minutes, it's very crappy but it does the job)
Beautiful intuitive proof!
Finally, I’ve been waiting for this!!
Very good explanation, very clear. Well done.
Started loving trig this year... it's so gorgeous! Anyways, since you mentioned it in the end, how would you go about generalizing this proof? Would love a follow-up video! Stay awesome :)
Flammable maths has a proof. Basically you define cos and sin as the real and imaginary parts of e^i(a+-b)
And go from there
Loved it! Thank you!
I can't believe it, you disappear for some time and when you come back you upload the video I needed! Magic isn't it?
Excellent video and explanation, greetings from Argentina.
You are the funniest mathematician I have ever seen master :)
Best explanation so far.
great video! that intro was epic. you definitely need a bloopers reel. :-) maybe dr. peyam could drop in and make mention of the "chen lu" too.
THANK YOU SO MUCH,
I loved this
That is amazing!
Very cool! I’ve always wanted to know how to find the formula is made rather than just memorising them
İ knew just trigonometric sum identities with complex analysis,in your issue i learnt new methods
That was splendid.
Nice geometric proof Steve. Another way to derive the double angle formulas is to start with euler's formula exp[i(a + b)] -> exp(ia) * exp(ib) -> ... and continue to reduce the form into the trig functions.
That's super nice!
Very good argument, it enough to persuade most people
This is like a hidden gem in youtube
I'm doing my English homework on a Robert Frost poem while watching a video about the proof of angle sum formulas for sine and cosine. Wow! I feel so smart right now.
Thank you very much your explanation is very good and simple
that's neat
AndDiracisHisProphet and cute lol
that too
Amazing and easy to understand
marvelous dear
Fantastic to get both in one rectangle.
AAAAAAAAA This is TOTALLY something I really need!! Trig identities are my Achilles's Heel in mathematics! Please please PLEASE make more of these!
Coincidentally do you know where I can look up trig identities problem sets I can try to work through? Or maybe make a few of your own for folks like me to practice with? :D
Calyo Delphi try khan academy
You're really a genius!!
Genius,,,, amazing method
This is beautiful
beautiful!
Thank you very much for the interesting video! I enjoyed it
My pleasure!!!!!!!!!!
That's a very clear explanation . Great thanks to you from egypt.
My pleasure!!!!
This is awesome.
Wow.! Love this channel for a very long time. That first 5 seconds where you're like 'da f***' had me rolling on the floor laughing. It's great to see mathematics treated like an adult by an adult. Not everything has to be apples, oranges and sterile PC banality. Thank you for making math more human and more alive.
Very nice to see the derivation in this way! (Altough I find it easier to use Euler's formula for complex exponentials to memorize these properties :D )
In2Mattle How?
cos(a+b) + i*sin(a+b) = exp(i*(a+b)) = exp(i*a)*exp(i*b) = (cos(a) + i*sin(a))*(cos(b) + i*sin(b))
= cos(a)*cos(b) - sin(a)*sin(b) + i*(sin(a)*cos(b) + cos(a)*sin(b))
=> cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b) and sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b)
I just watched the bprp's video on this topic. How couldn't I see it! Thank you for your answer anyway!
Andy Arteaga yea i put that in the description and the endscreen after I read this thread
Yeah that an interesting proof too. But the trig one gives a more direct sense of the physicality of the situation and uses trig directly. @@In2Mattle
Heyy nice proof i really appreciate!
Can you make the integral from 0 to infinity of sin(x)/x by laplace transform? thank you
I always used that formula but now I also know how it is derived
Thank you very much.
I am glad!!! yay
Really nice!
wow, thank you, i´m startled
Flawless.
Thank you!
Wow! Finally! A new video!
Great video!
This proof is elegant
"And as usual, thats it" what a perfect catchfrase
Nice explanation 😃
One of the best videos i have ever watched. I didnt understood the proof in plane trigonometry (sl loney) but this proof was awsome. Btw can we proof sin(a-b) and cos(a-b) with same method with little chamges?
No one is perfect , well said
this tools are very useful when either one angle is unknown but if both were given then u can forget this equation since the calculator can simplify the calculation of both sine and cosine . just add them two angles and insert it into the function
Thank u so much
U are awesome👏👏👏
This was beautifull. I remember the proof in my students book be like super confusing.
Wow sir 😯❣️😍 keep growing sir
Mathematics is magical subject 💗❣️💣
Lovely...and i dare to say that it can be easily generalized
what took so long for this video...
great job.
i love your videos...
Rotation matrix R[a] = anticlockwise rotation by angle a and is the matrix [[cos a -sin a] [sin a cos a]]
to rotate anticlockwise by angle a+b, R[a+b] = R[a].R[b]. This will give formulæ for sin(a+b) and cos(a+b) in terms of sin a, cos a, sin b and cos b.
Likewise eᴵª . eᴵᵇ = (cos a + i sin a)(cos b + i sin b) = eᴵ⁽ª⁺ᵇ⁾ = cos(a+b) + i sin(a+b) will give these formulæ too.
However, none of the above is taught until the elementary trig course is complete. The method you give is the best I've seen.
Cool, I like it
Fabulous 👌
What's cool also is that you can use the picture to show that if a