Proof of angle addition formula for sine | Trigonometry | Khan Academy
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- เผยแพร่เมื่อ 23 ส.ค. 2024
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Loved every time he said you should be getting excited
Thanks my friend, you're getting me through uni 👌
Shouldn't you justify the construction of the diagram (@ 3:19) before going through the proof so that your reasoning doesn't appear to have come from observing the construction but rather from definitions?
Maybe someone can correct me if I'm misguided here, but I think Sal is attempting to illustrate the geometric reasoning behind it so people who are estranged from rigorous mathematical definitions can get an intuition for this. Though I'm not versed in math history, it's my understanding that the early pioneers of trigonometry were using physical objects as their sandbox to construct these ideas.
You are correct, and I struggled with this for a while. The construction can be justified by the ability to create a right triangle of arbitrary hypotenuse. This can be thought of in terms of rays that can extend from an arbitrary angle, and then drop down a line to form a right triangle. So create one of these with hypotenuse 1, then one of them with hypotenuse equal to the base leg (cos b). Then you have the construction.
Note that this proof only covers Acute angle addition. For Obtuse angle addition, you'd need something like Ptolmey's Theorem to prove it using Chords.
@@urthogie well couldn't you draw an equivalent diagram for obtuse angles?
6:45 SO THAT'S WHY!
hehe. Great video thanks! always wondered about this property
Love every time he said you might want to try it .
Literally all I saw in class was this irregular drawing and came here now I get it bless up
From fig 0:17 trigonometrically >
Sinxcosy + cosxsiny
( ed=sinxcosy and ef=cosysiny)
Oh mo gosh that was sooo easy thank you for enlightening my mind Mr. Sal
sal is the best teacher like if you agree
3:42 I thought he was gna say so assuming you gave up ...
CAN anyone please tell me how AD is equal to 1, what if other values are taken??
Wonderful clear explanation, thank you!
Will sin(x+y) always form 90 degree triangles that are stacked on top of each other or could the two triangles rest on the x axis?
It doesn't matter literally. The 90 degree angle triangles is to help you visualize the relationship between
Sin(x+y)
And
Sinxcosy + cosxsiny
This is visualization but in actuality you're just using this expression to solve equations. It's like sin cos tan they tie in the relationship of the triangle sides, same to this. Just an identity
Thank you sir,this video really opened my mind 😁
Oh no. I have only watched the first 30 seconds but I think this video is going to explain to me something I was trying to understand through hours of reading. I'll reply to myself with an update if this does in fact take place.
Wow, you did it! Thanks for the video! I understand now! I should have come here first! Thanks again!
Yes, me too!!! Great explanation! It's so so wonderful when there is a clear explanation. A lot of math is like this, just needs a clear well presented explanation that doesn't leave any steps out. I had forgotten about the tranverse through two parallel lines and how angles are the same.
@@ramyhuber8392 that's awesome! I totally agree
I've been wondering how and why the formula works and now I know and I wonder what the complex number proof of this would be.
Yeah and the teachers never explain how the formula came
They just say learn it, it is important
Thank you man. I'm gonna show it to my students
I wish i could upvote this more than 1000 times
Thank you so much for your best explanation. ❤❤
How woudl proof of x+y>90 degrees look like?
That's my question
Thank you so much for the lesson
this hypothenus from side AD, is that equal to 1 because it is on a unit circle?
It's not 1 for any particular reason, really. It's an assumption that makes the ensuing proof easier. One could assume it's equal to any quantity; it's easiest to just set it equal to 1.
but what if the length of AD was not 1?
thank you Sir 😍
Thank you!!
thank you so much!
I have not understood how did you why you said to get triangle DCE is equal to 90_y
สุดยอดครับ😊😊
i have been trying to figure out how to multiply lengths so you get the length of the product and then trying to use euclidean geometrical proof to do this and then i see this proof.
thank you it was helpful
not sure if anyone will get back to me in time-but why do we make this diagram
thank you sir
If x/y are the angles of right angled triangle they should be always acute or equal to 90 degrees
But why should we care about sin 160, sin 180, and all.....
Please clear it quickly as i have paused their
After 4 decades plus since trig, I still have not seen anyone explain why the 2 angles X and Y are not simply added together, and then find the sine from a trig table or calculator, instead of using this cumbersome formula.
Because the pioneers who developed those trig tables used these formulas to gain more values for their tables. The easy to extract information first was for degrees of 0,30,45,60,90. From here subtracting or adding these numbers up using the formulas will give an exact answer for sin(45+30) =sin45cos30+cos45sin30=sin(75)
This formula and several other ingenious methods of approximating values of sin led to the huge tables of values of sin up to great precision
@@robonthecob5092 Calculators are available. These formulas are simply a waste of time and brain power.
ik im late but its to find exact values for them@@lwh7301
I am Bangladeshi . I am seeing your tech.
Managed to get up to sin(x+y)=DE + cosx siny, but couldn't figure out how to express DE. I'm a bit suspicious of the right triangle DCE, perhaps its angles have some relation to angle x, but i can't see it.. finished watching, now i get it. ha
why is AC not equal to one? the hypo stays the stays the same on the unit circle are these triangles different ratio of lengths then?
Great question
AC = 1 if and only if x=0. Otherwise AC=cos(x) by assuming AD=1
MathProofsable Actually I think that’s only true if AB remains constant; AC an be 1 even if x>0 by increasing the length of AB
@@iaexo AB is dependent on AC which is in turn dependent on AD. So once we fix AD=1, AC is determined by the construction of the right angle at C and then AB is determined by the construction of the right angle at B.
@@MathProofsable Ah yes I see now thanks
this is great, thanks :)
Is segment AD=AC because they are both hypothenuse
Nice!
Surprisingly, 1.25 times speed seems abnormally normal
good
Why isn't AC=1 since it is also a radius of a unit circle?
he assumed only AD equal to 1
because it isn't
Insane~~
mind = blown
I love proofs
Tryhard
Can someone explain me how he got that the angle EDC is y ?
T4l0nITA total angle of a right angle triangle(or any triangle is 180°)
The angles given on the triangle are 90 and 90-y.
One more angle left so let's do the math.
90 (first angle)
90-y (second angle)
(We don't know the third angle so let's just put 'm' into it and use the substitution method) ( to avoid confusion just draw out the triangle onto another piece of paper and fit it back in once you're done)
m (third angle)
180= 90+90-y+m
180-90-90=-y+m
0=-y+m
y=m
So, basically put the pieces together and you would get that it is y.
Radio
can anyone tell me that why we have to assume that the hypotneuse would be 1 only.............. can anyone justify me we are taking one just to make calc esier...............................
The hypotenuses don't have to be equal to one. They're arbitrarily equal to one for simplicity. If, however, you choose to have a hypotenuse of length "r", then you have to multiply the hypotenuse in every trig. ratio by its respective "r". For example, cos(θ)=x/r, where "r" is the hypotenuse. When "r" is equal to one, cos(θ)=x/r becomes cos(θ)=x. The reason why the value of "r" doesn't really matter is because we're dealing with ratios that don't change.
how to remember this?
unfortunately this doesn't help you to memorize the formula magically. I think it just clears the non-senseness of the rule
If AD is 1 then why can't AC also 1?
Cause it ain’t bor cos x = 1 only when x=0
population
yes
why is AD =1?
It is just so that the calculation would look a bit cleaner... even if length of AD is given any arbitrary variable (for example, h), you would get the same result. Try it yourself!
it's because of the unit circle, the radius is one. since AD is a radius as well, it is equal to one. hope that helps :)
AC=cos(x)*AD and DC=sin(x)*AD, so letting AD lets the calculations be much cleaner.
What i don't like about this proof is how DA is purposefuly = 1 to make the demonstration simpler. The real proof must involve a generic DA even though you get a lot of divisions.
Victor Serra since you can scale triangles proving it for 1 suffices
Too complex for you huh?
Victor Serra The real proof doesn't require radii greater than one. The trigonometric functions are based upon the the unit circle.
all these are inside a unit circle
ad is radius
How is AD =1?
By assumption.
it doesn't change the result, eventually ratio would be still sin(x+y).
The proof is done in a unit circle.