Trigonometry Identity Review/Fun Revisiting the proofs of some trigonometry identities.
Trigonometry Identity Review/Fun
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- i've already made a handful of videos that covers what
- I'm going to cover, the trigonometric identities I'm
- going to cover in this video.
- The reason why I'm doing it is that I'm in need of review
- myself because I was doing some calculus problems that required
- me to know this, and I have better recording software now
- so I thought two birds with one stone, let me rerecord a
- video and kind of refresh things in my own mind.
- So the trig identities that I'm going to assume that we know
- because I've already made videos on them and they're a
- little bit involved to remember or to prove, are that the sine
- of a plus b is equal to the sine of a times the cosine
- of b plus the sine of b times the cosine of a.
- That's the first one, I assume, going into this video we know.
- And then if we wanted to know the sine of-- well, I'll just
- write it a little differently.
- What if I wanted to figure out the sine of a plus-- I'll
- write it this way-- minus c?
- Which is the same thing as a minus c, right?
- Well, we could just use this formula up here to say well,
- that's equal to the sine of a times the cosine of minus c
- plus the sine of minus c times the cosine of a.
- And we know, and I guess this is another assumption that
- we're going to have to have going into this video, that the
- cosine of minus c is equal to just the cosine of c.
- That the cosine is an even function.
- And you could look at that by looking at the graph of the
- cosine function, or even at the unit circle itself.
- And that the sine is an odd function.
- That the sine of minus c is actually equal
- to minus sine of c.
- So we can use both of that information to rewrite the
- second line up here to say that the sine of a minus c is equal
- to the sine of a times the cosine of c.
- Because cosine of minus c is the same thing
- as the cosine of c.
- Times the cosine of c.
- And then, minus the sine of c.
- Instead of writing this, I could write this.
- Minus the sine of c times the cosine of a.
- So that we kind of pseudo proved this by knowing this
- and this ahead of time.
- Fair enough.
- And I'm going to use all of these to kind of prove a bunch
- of more trig identities that I'm going to need.
- So the other trig identity is that the cosine of a plus b is
- equal to the cosine of a-- you don't mix up the cosines and
- the sines in this situation.
- Cosine of a times the sine of b.
- And this is minus-- well, sorry.
- I just said you don't mix it up and then I mixed them up.
- Times the cosine of b minus sine of a times the sine of b.
- Now, if you wanted to know what the cosine of a minus b is,
- well, you use these same properties.
- Cosine of minus b, that's still going to be cosine on b.
- So that's going to be the cosine of a times the cosine--
- cosine of minus b is the same thing as cosine of b.
- But here you're going to have sine of minus b, which is the
- same thing as the minus sine of b.
- And that minus will cancel that out, so it'll be plus sine
- of a times the sine of b.
- So it's a little tricky.
- When you have a plus sign here you get a minus there.
- When you don't minus sign there, you get
- a plus sign there.
- But fair enough.
- I don't want to dwell on that too much because we have many
- more identities to show.
- So what if I wanted an identity for let's
- say, the cosine of 2a?
- So the cosine of 2a.
- Well that's just the same thing as the cosine of a plus a.
- And then we could use this formula right up here.
- If my second a is just my b, then this is just equal to
- cosine of a times the cosine of a minus the sine of
- a times the sine of a.
- My b is also an a in this situation, which I could
- rewrite as, this is equal to the cosine squared of a.
- I just wrote cosine of a times itself twice or times itself.
- Minus sine squared of a.
- This is one I guess identity already.
- Cosine of 2a is equal to the cosine squared of a minus
- the sine squared of a.
- Let me box off my identities that we're showing
- in this video.
- So I just showed you that one.
- What if I'm not satisfied?
- What if I just want it in terms of cosines?
- Well, we could break out the unit circle definition
- of our trig functions.
- This is kind of the most fundamental identity.
- The sine squared of a plus the cosine squared
- of a is equal to 1.
- Or you could write that-- let me think of the
- best way to do this.
- You could write that the sine squared of a is equal to
- 1 minus the cosine sign squared of a.
- And then we could take this and substitute it right here.
- So we could rewrite this identity as being equal to the
- cosine squared of a minus the sine squared of a.
- But the sine squared of a is this right there.
- So minus-- I'll do it in a different color.
- Minus 1 minus cosine squared of a.
- That's what I just substituted for the sine squared of a.
- And so this is equal to the cosine squared of a minus 1
- plus the cosine squared of a.
- Which is equal to-- we're just adding.
- I'll just continue on the right.
- We have 1 cosine squared of a plus another cosine squared
- of a, so it's 2 cosine squared of a minus 1.
- And all of that is equal to cosine of 2a.
- Now what if I wanted to get an identity that gave me
- what cosine squared of a is in terms of this?
- Well we could just solve for that.
- If we add 1 to both sides of this equation, actually,
- let me write this.
- This is one of our other identities.
- But if we add 1 to both sides of that equation we get 2 times
- the cosine squared of a is equal to cosine of 2a plus 1.
- And if we divide both sides of this by 2 we get the cosine
- squared of a is equal to 1/2-- now we could rearrange these
- just to do it-- times 1 plus the cosine of 2a.
- And we're done.
- And we have another identity.
- Cosine squared of a, sometimes it's called the power reduction
- identity right there.
- Now what if we wanted something in terms of
- the sine squared of a?
- Well then maybe we could go back up here and we know from
- this identity that the sine squared of a is equal to 1
- minus cosine squared of a.
- Or we could have gone the other way.
- We could have subtracted sine squared of a from both sides
- and we could have gotten-- let me go down there.
- If I subtracted sine squared of a from both sides you could get
- cosine squared of a is equal to 1 minus sine squared of a.
- And then we could go back into this formula right up here and
- we could write down-- and I'll do it in this blue color.
- We could write down that the cosine of 2a is equal to--
- instead of writing a cosine squared of a, I'll write this-
- is equal to 1 minus sine squared of a minus
- sine squared of a.
- So my cosine of 2a is equal to?
- Let's see.
- I have a minus sine squared of a minus another
- sine squared of a.
- So I have 1 minus 2 sine squared of a.
- So here's another identity.
- Another way to write my cosine of 2a.
- We're discovering a lot of ways to write our cosine of 2a.
- Now if we wanted to solve for sine squared of 2a we could
- add it to both sides of the equation.
- So let me do that and I'll just write it here for
- the sake of saving space.
- Let me scroll down a little bit.
- So I'm going to go here.
- If I just add 2 sine squared of a to both sides of this, I
- get 2 sine squared of a plus cosine of 2a is equal to 1.
- Subtract cosine of 2a from both sides.
- You get 2 sine squared of a is equal to 1 minus cosine of 2a.
- Then you divide both sides of this by 2 and you get sine
- squared of a is equal to 1/2 times 1 minus cosine of 2a.
- And we have our other discovery I guess we could call it.
- Our finding.
- And it's interesting.
- It's always interesting to look at the symmetry.
- Cosine squared-- they're identical except for you have a
- plus 2a here for the cosine squared and you have a minus
- cosine of 2a here for the sine squared.
- So we've already found a lot of interesting things.
- Let's see if we can do anything with the sine of 2a.
- Let me pick a new color here that I haven't used.
- Well, I've pretty much used all my colors.
- So if I want to figure out the sine of 2a, this is equal
- to the sine of a plus a.
- Which is equal to the sine of a times the co-- well, I don't
- want to make it that thick.
- Times the cosine of a plus-- and this cosine of a,
- that's the second a.
- Actually, you could view it that way.
- Plus the sine-- I'm just using the sine of a plus b.
- Plus the sine of the second a times the
- cosine of the first a.
- I just wrote the same thing twice, so this is just people
- to 2 sine of a, cosine of a.
- That was a little bit easier.
- So sine of 2a is equal to that.
- So that's another result.
- I know I'm a little bit tired by playing with all of
- these sine and cosines.
- And I was able to get all the results that I needed for my
- calculus problem, so hopefully this was a good review for
- you because it was a good review for me.
- You can write these things down.
- You can memorize them if you want, but the really important
- take away is to realize that you really can derive all of
- these formulas really from these initial formulas
- that we just had.
- And even these, I also have proofs to show you how to get
- these from just the basic definitions of your
- trig functions.
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