Trigonometric identities
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Trigonometric Identities
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Pythagorean identities
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Proof: sin(a+b) = (cos a)(sin b) + (sin a)(cos b)
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Proof: cos(a+b) = (cos a)(cos b)-(sin a)(sin b)
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Trig identities part 2 (part 4 if you watch the proofs)
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Trig identies part 3 (part 5 if you watch the proofs)
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Addition and subtraction trig identities
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Law of cosines
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Law of cosines
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Navigation Word Problem
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Proof: Law of Sines
Proof: cos(a+b) = (cos a)(cos b)-(sin a)(sin b) Proof of the trig identity: cos(a+b) = (cos a)(cos b)-(sin a)(sin b)
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- Welcome back.
- We'll now try to see what trigonometric identity we can
- come up with if we start off with cosine of alpha plus beta.
- Let's see if we can rewrite this as another combination
- of cosines and sines of alpha and beta.
- So let's get started.
- And if you've already watched the sine equivalent of this,
- this proof will be pretty similar in how we operate.
- And we get a kind of similar answer.
- And something interesting is to kind of compare the difference
- between sine of alpha plus beta and cosine of alpha plus
- beta after we're done.
- So just like that last proof, let's say that this angle-- no,
- that color isn't bright enough.
- Let's do yellow.
- Let's say that this angle right here is alpha, and this
- angle right here is beta.
- Right?
- We want to know cosine of alpha plus beta.
- So alpha plus beta is this large angle right here.
- Right?
- So what's the cosine of that?
- SOH, CAH, TOA.
- So cosine is adjacent over hypotenuse.
- Right?
- SOH, CAH, TOA.
- CAH.
- So cosine is equal to adjacent over hypotenuse.
- So for this large angle, what's the adjacent?
- It's line AC.
- So that equals-- line AC, that's the adjacent.
- The length of line AC over the length of line--
- what's the hypotenuse?
- AB, right?
- All right.
- AC over AB.
- Now let's see what we can do with this.
- AC-- adjacent over hypotenuse.
- Now can we write AC in any other interesting way-- a
- combination of some of the other lines on this very
- fortunately designed graph?
- Let's see.
- Well isn't AC the same thing as AF, this big line,
- minus-- what is this?
- This is a D, right?
- OK.
- That's a D.
- Let me rewrite that as a D.
- D as in dog.
- There you go.
- So AF minus DE.
- Right?
- Oh, I forgot to draw some things.
- We assume that this line is perpendicular to that line.
- We assume this line is perpendicular to that line.
- We assume that line is perpendicular to that line.
- Right?
- And then by definition, that is because we drew it that way.
- But anyway.
- So now you know that this line is parallel to this line and
- this line is perpendicular.
- So we know that AF, this long line, minus DE is equal to AC.
- Does that make sense?
- AF, this big line, minus the shorter line is
- the same thing as AC.
- Right?
- So let me write that down.
- That equals AF minus DE, all of that over AB.
- And then, of course, we can rewrite that as-- and I'm going
- to switch to some different colors-- as AF over AB.
- Let me switch to maybe green.
- Minus DE over AB.
- So we have now AF over AB minus DE over AB.
- And those are kind of nonsensical ratios to me.
- Wouldn't it be great if we could express it
- somehow as AF over AE?
- Because then we could say well that's cosine of alpha, and
- do something from there.
- Well let's try.
- So let's try to rewrite this first expression.
- So I'll switch back to the purple just so you
- know where this first expression is coming from.
- Let's see if we can break this down as AF over AE
- times something else.
- Well, we could just algebraically do it.
- That's equivalent to AF over-- I might run out of space--
- over AE times AE over AB.
- And you're saying, Sal where did you get that from?
- Well, you can kind of say my motivation was to have AF
- as a ratio of over AE.
- And I just set it up so that the multiplication would cancel
- out, because the AE's would cancel out and you'd be
- left with AF over AB.
- Right?
- So this is a reasonable thing to do.
- I hope you see.
- And let me switch to the green and do something similar.
- DE over AB doesn't make much sense.
- But if I could maybe do DE over BE, then if this angle I can
- see is similar to alpha or beta then maybe I can
- make some progress.
- So let's say DE over BE times-- and we'll do the same thing.
- You just have to multiply times BE over AB.
- And just like in that sine proof, and we'll do the same
- thing here, let's figure out what this angle up here is.
- Right?
- Because if we know that then these ratios become useful.
- So if this angle is an alpha, then we know that this
- angle right here is alpha.
- Right?
- Because DE-- because this line-- is parallel to AF.
- You learned that in geometry.
- And if this angle is alpha we know that this angle right here
- is 90 minus alpha, because it's complementary.
- Right?
- Because this whole angle is 90 degrees, so this
- is 90 minus alpha.
- And since this angle 90 minus alpha, this angle 90, and this
- angle add up to 180, we could figure out that this is alpha.
- And if you don't believe me add up alpha plus 90 plus 90 minus
- alpha, and you will get 180 degrees.
- So this angle up here, angle DBE, is alpha.
- So that's very interesting.
- So can we rewrite these ratios as the sines or cosines
- of alpha's or beta's?
- Well, let's try.
- Let me switch back to purple.
- So that equals-- what is AF over AE?
- Well if we look at this right triangle, that's the adjacent
- over the hypotenuse for alpha.
- Right?
- Adjacent over hypotenuse, that's cosine.
- So it's cosine of alpha.
- And what's AE over AB?
- Well, they're similar.
- If we look at this big right triangle right here that is the
- adjacent over the hypotenuse for beta, so it's
- cosine of beta.
- Switch my colors.
- Minus DE over BE.
- Well this is alpha, right?
- Now there's a little smudge.
- You probably can't read it.
- But that was alpha.
- We showed that that was alpha.
- So DE is the opposite and BE is the hypotenuse.
- Opposite over hypotenuse is sine, right?
- So that's sine of alpha.
- And what is BE over AB?
- Look at this triangle again.
- Well, for beta that is BE is the opposite and
- AB is the hypotenuse.
- So opposite over hypotenuse for beta.
- So it's the sine of beta.
- Times the sine of beta.
- I'm running out of space.
- I have to go to another line.
- Pretty neat.
- I'll rewrite everything in a new and exciting color.
- OK.
- Let me do it in light blue.
- So we now know that the cosine of alpha plus beta is equal to
- the cosine of both of them multiplied-- so cosine of
- alpha, cosine of beta-- minus the sine of both of
- them multiplied.
- Minus sine of alpha times sine of beta.
- I hope you found that as satisfying as I do.
- See you in the next presentation.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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