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
Trigonometric Identities Introduction to trigonometric identities
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- Welcome back.
- I'm now going to do a series of videos on the
- trigonometric identities.
- So let's just review what we already know about the
- trig function, so let me just write SOHCAHTOA.
- That tells us, and we've actually extended this with the
- unit circle definition, but if you watch those videos, you'll
- realize that the unit circle definition directly
- uses SOHCAHTOA.
- So we'll just stick with SOHCAHTOA because I think it'll
- help make some of what we're about to do seem a little bit
- more straightforward and will kind of verge on the unit
- circle definition anyway.
- So we know that sine of theta is equal to opposite
- over hypotenuse, right?
- So cosine of theta is equal to adjacent over hypotenuse, and
- then the tangent of theta is equal to opposite
- over adjacent.
- So let's draw that out on a right triangle.
- We could do this with the unit circle as well, and
- it would make sense.
- Let's see if we can find a relationship between sine,
- cosine and tangent.
- There's my right triangle.
- Let's call this theta.
- This is the hypotenuse h.
- This is the opposite side, right, opposite of theta.
- This is theta right here.
- This is the adjacent side, right?
- Well, what do we know about the relationship between
- the opposite adjacent side and then the hypotenuse?
- What does the Pythagorean theorem tell us?
- Oh, yeah, this side squared plus this side squared is equal
- to the hypotenuse squared, so we could write that down.
- a squared plus o squared is equal to the hypotenuse
- squared, right?
- And then this is just an equation, so if we want to, we
- could divide both sides of this equation by h squared,
- and so what do we get?
- We get a squared over h squared plus o squared over h squared
- is equal to 1, right?
- And then I could rewrite that as a over h squared plus o
- over h squared is equal to 1.
- Now, do these look at all familiar?
- Well, we have them here, right?
- This is a over h, this is o over h, so we could
- just substitute.
- So this is just cosine of theta squared.
- And this is how you write cosine squared.
- You could put a parentheses around the whole thing and then
- square it, but this is just the notation people use.
- Plus opposite over adjacent squared, so that's sine theta
- squared is equal to 1.
- So that's our first trig identity.
- So if you know the sine of theta, it's very easy to figure
- out the cosine of theta, right?
- You could just solve this equation.
- If I know that the-- I don't know.
- Let's say I know that the sine of theta is 1/2, right?
- Then what is the cosine of theta?
- The cosine of theta is what?
- Well, I know the sine of theta is 1/2, right?
- So I would say cosine squared of theta plus sine of theta
- is 1/2, so 1/2 squared is equal to 1, right?
- So cosine squared theta plus 1/4 is equal to 1.
- So we have cosine squared theta is equal to 3/4, or cosine of
- theta would be the square root of this, right?
- We just take the square root of both sides.
- It would be the square root of 3/2.
- And you probably remember that from our whole presentation
- on the 30-60-90 triangle.
- So I just wanted to show you a use of this trig identity
- that's usually written sine squared plus cosine
- squared is equal to 1.
- So let's extend that one a little bit.
- Let's just play with the ratios and see what else we can--
- other identities we can discover.
- Whoops!
- Clear image, invert colors.
- So we know that sine squared theta plus cosine squared
- theta is equal to 1.
- The one thing we could do is we could divide both sides of this
- equation by cosine squared of theta, and let's just see what
- happens when we do that.
- So if we say cosine squared theta, right?
- You have to distribute across both terms.
- Cosine squared of theta, and then cosine squared of theta.
- Well, what's sine squared theta over cosine squared theta?
- That's the same thing as sine of theta over cosine of theta
- squared plus this is 1 over cosine theta squared, right?
- I mean, 1 squared is 1, so I just rewrote it.
- So sine over cosine theta, I think we learned that already.
- That's just the tangent of theta.
- And in case you actually haven't learned that already,
- think about it this way.
- Sine is opposite over the hypotenuse, right?
- So that's opposite over hypotenuse.
- And then cosine is adjacent over hypotenuse.
- So adjacent over hypotenuse.
- So then that equals opposite over hypotenuse times
- hypotenuse over adjacent, right?
- Just dividing by a fraction is the same thing as multiplying
- by its reciprocal.
- That's all I did.
- And that equals opposite over adjacent, right?
- So that just says sine of theta over cosine of theta is
- equal to tangent of theta.
- So sine squared theta over cosine squared theta is tan
- squared theta, then plus 1 is 1 over cosine theta squared.
- And now I'm going to introduce a new trig ratio, it's really
- just 1 over cosine theta.
- So 1 over cosine theta-- and I'm going to summarize this at
- the end, just so it's not too confusing-- is actually
- the secant of theta.
- And this is just another ratio, right?
- The secant of theta, instead of being the adjacent over the
- hypotenuse, would be the hypotenuse over the
- adjacent, right?
- It's just 1 over cosine theta.
- Nothing fancy here.
- So secant of theta.
- So that equals secant squared of theta.
- I know it can be a little overwhelming initially, just
- because I'm, you know, throwing out all these new terms, secant
- is 1 over cosine theta, but once you just play around with
- these enough and get familiar with the terms, it'll make
- sense, and it'll be a little more natural to you.
- So this could be-- you could view this as
- another trig identity.
- And actually, I don't even remember if I've
- taught it already.
- I mean, you could view this as a trig identity, although
- that's almost definitional.
- And then, of course, you can-- in case I haven't done it
- already, you now know that sine of theta over cosine of theta
- is equal to tangent of theta.
- And that's right here with, I guess you could
- say, the proof of it.
- So let me keep introducing you to more things, and if this is
- really daunting, maybe you just can rewatch it, and
- hopefully, it'll make sense.
- Let me see, clear image.
- So what have we learned so far?
- We learned that sine squared theta plus cosine squared
- theta is equal to 1.
- We learned that sine of theta over cosine of theta is
- equal to tangent of theta.
- We learned that the tangent squared of theta plus 1 is
- equal to the secant of theta.
- And here, let me actually write this definition down.
- The secant of theta-- oops, is equal to the secant
- squared of theta, sorry.
- And the secant of theta is just 1 over cosine of theta.
- This is something you really should just memorize, that
- secant is 1 over cosine.
- And if you're wondering what 1 over sine is, 1 over sine of
- theta, it's the cosecant-- the abbreviation is csc-- of theta.
- And if you're wondering what 1 over the tangent is,
- it's the cotangent.
- And you just might want to memorize these.
- And this often confuses me, that 1 over the cosine is the
- secant, but 1 over the sine is the cosecant, so it's kind of
- almost the opposite, right?
- 1 over the sine has a co in it, while 1 over the cosine
- doesn't have the co in it.
- So that might help you remember things.
- So I think that's all I have time for now.
- In the next presentation, I'm going to introduce you to a
- couple more trig identities.
- See you soon.
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