Proof: sin(a+b) = (cos a)(sin b) + (sin a)(cos b) Proof of the trig identity sin(a+b) = (cos a)(sin b) + (sin a)(cos b)
Proof: sin(a+b) = (cos a)(sin b) + (sin a)(cos b)
⇐ Use this menu to view and help create subtitles for this video in many different languages. You'll probably want to hide YouTube's captions if using these subtitles.
- Welcome back.
- I'm now going to do a proof of a trig identity, which
- I think is pretty amazing.
- Although, I think, the proof isn't that obvious.
- And I'll have to admit ahead of time, this isn't something that
- would have occurred to me naturally.
- I wouldn't have naturally drawn this figure just
- to start off with.
- Let's just say we want to figure out some other way to
- write the sine of alpha plus beta, where alpha and beta are
- let's say, two separate angles.
- So if I had the sine of 40 and 50 degrees, I'd want to know--
- this would obviously be the sine of 90, which is easy.
- But could I rewrite that as some combination of the sine
- of 40 and the sine of 50 or whatever?
- I think you'll see where this is going.
- So let's go back to this diagram and let's say
- that this-- let me pick a better color.
- Let's say that this is angle alpha and this is angle beta.
- Than this whole angle right here is angle alpha plus beta.
- So we want to figure out the sine of alpha plus beta.
- Well, the sine of alpha plus beta, the sine of this
- whole angle, opposite over hypotenuse.
- Opposite this whole angle is if we use this right angle-- or
- this right triangle, triangle BAC.
- Opposite is BC, so that equals BC.
- I'll draw a little line over it.
- BC over the hypotenuse, AB.
- BC over AB is the sine of alpha plus beta.
- Well, can be write BC over AB differently?
- Let's see if we can.
- And probably, the person who first figured out this proof
- was just playing around.
- They drew this diagram, they said, can I write
- BC any differently?
- Well BC-- this whole length-- is the sum of BD and EF.
- And we know that because this is a horizontal line right now
- and you can figure that out just by looking at all
- the right angles.
- But this is a horizontal line.
- So BC is the same thing is BD plus EF.
- Let's write that one down.
- BC is the same thing as BD plus EF.
- And then still, all of that, over AB.
- All I did is I rewrote BC as a sum of this segment and this
- segment, which should make sense to you, hopefully.
- And then we can of course, rewrite that as equal to BD
- over AB plus EF over AB.
- So BD over AB plus EF over AB.
- And these are kind of nonsensical ratios, right?
- BD over AB, what can I do with that?
- And EF over AB, what can I do with that?
- Wouldn't it be more interesting if I could do like BD over BE.
- That'd be an interesting ratio because that would be a
- segment over its hypotenuse.
- So let's see if we can rewrite it somehow like that.
- Well, we could just do it mathematically.
- We could say this is equal to BD over BE times BE over AB.
- So this might seem non-intuitive to you, but
- it kind of makes sense.
- We didn't pick BE completely arbitrarily.
- We said we know what BD is, so let me pick another side that I
- can do something maybe with real trig ratios.
- And so I said BD over BE times BE over AB is
- equal to BD over AB.
- I hope I don't confuse you with all these letters.
- But that makes sense, right?
- Because these two terms would just cancel out.
- If we're just multiplying these fractions then you would
- get back to this top term.
- Let me actually make sure that you understand
- that this-- whoops.
- That this term and this term are the same thing.
- And now let's do that second term.
- We know EF, wouldn't it be good if we could relate EF to
- something, like it's the hypotenuse of this
- right triangle?
- Like AE.
- So let's do that.
- So let's put the plus sign there.
- EF over AB is the same thing as EF over AE times AE over AB.
- Once again, we're just multiplying fractions.
- These would cancel out and you would get this again.
- Let me make sure you understand that this term is the
- same thing as this term.
- And you can just multiple out the fractions and that's
- what you would get.
- Now before we progress with this whole line of
- thought that we're doing.
- Let's see if we could figure out something else interesting
- about this strange set of triangles and shapes
- that I've drawn.
- It's actually pretty neat.
- IF this angle is alpha-- we have line AF.
- EF is perpendicular to it, right?
- And DE is perpendicular to EF, right?
- So DE, this line, and AF are parallel.
- Since AF is parallel to DE and then, AE intersects both,
- we know that, what is that?
- The inner angles?
- Yeah, I think that's called inner angles
- with parallel lines.
- That this is also equal to alpha.
- You can imagine long parallel line here, long parallel here,
- and then this line intersects both.
- So if this is a little confusing maybe you want to
- review a little bit of the parallel line geometry, but I
- think this might make sense.
- So if this angle is alpha, then this angle right here
- is complementary to it.
- So it's 90 minus alpha.
- And if this angle is 90 minus alpha, this
- angle is obviously 90.
- Then we know that this angle plus this angle plus this
- angle has to equal 180.
- So we know that this is equal to alpha.
- If that doesn't make sense to you, think about this: alpha
- plus 90 minus alpha plus 90-- that's a minus.
- Minus alpha.
- Plus 90 is what?
- Alpha plus 90 minus alpha.
- So this minus alpha and alpha cancel out and you just have 90
- plus 90 and that equals 180.
- So we know that this angle right here, I know it's
- getting really small and probably hard to read.
- We know that this angle here is alpha.
- So let's get back to what we were progressing,
- what we were doing here.
- So what is BD over BE?
- BD over BE.
- Well, that's the adjacent to this alpha, which is
- the same angle really.
- BD over BE, so it's adjacent over hypotenuse.
- So that is equal to the cosine of alpha.
- And what's BE over AB?
- Well, if we look at this larger right triangle, that is the
- opposite of beta times its hypotenuse.
- So what's opposite over hypotenuse?
- S O H.
- So sine of beta is BE over AB.
- So this is sine of beta.
- And now let me switch to magenta.
- What's EF over AE?
- If we look at this right triangle right here,
- is opposite over hypotenuse for alpha.
- So it's sine of alpha.
- Opposite over hypotenuse.
- And what's AE over AB?
- So now we're looking at this large right triangle here.
- AE over AB.
- Well, that's the adjacent of beta over the hypotenuse.
- Well, what's adjacent over hypotenuse?
- That's the cosine.
- Cosine of beta, of this beta right here.
- I think we're done.
- This is to me, fairly mind blowing.
- That the sine of alpha plus beta is equal to the cosine of
- alpha times the sine of beta.
- Plus the sine of alpha times the cosine of beta.
- What's neat about this is that it kind of came out of this
- nice symmetric formula.
- It's not this big, hairy thing.
- You might have even guessed it.
- I don't know.
- I just find it very neat.
- We went through this big convoluted proof with this big
- convoluted shape, but we got this nice symmetric trig
- identity out of it.
- So hopefully you found that amazing as well and in the next
- presentation I'll do a proof for cosine of alpha plus beta.
- See you soon.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
Have something that's not a question about this content?
This discussion area is not meant for answering homework questions.
Share a tip
When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
- disrespectful or offensive
- an advertisement
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site