Trig identies part 3 (part 5 if you watch the proofs) Continuation of the playing around with trig identities
Trig identies part 3 (part 5 if you watch the proofs)
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- Welcome back, because I was hitting against the
- YouTube 10-minute limit.
- But all I was saying is, we said, you know,
- cosine of minus a.
- So I drew a right triangle with a, and then I showed you minus
- a, and I said, well, all of the lengths are going to be the
- same, but now the direction of-- and we're kind of assuming
- this is all on the unit circle.
- If you don't remember the unit circle, maybe you'll want to
- rewatch the videos we have on that.
- But I'm just showing you that the cosine of minus a is equal
- to this side over the hypotenuse, and this hypotenuse
- is the same as this hypotenuse, right?
- So cosine of minus a is adjacent over this hypotenuse,
- while cosine of a is adjacent over this hypotenuse.
- But it's the same thing, so we know that cosine of minus
- a is equal to cosine of a.
- Actually, by definition, that makes it a-- I don't want to
- confuse you too much, but that makes cosine an even function,
- and I'll show you more.
- Actually, I should do a whole presentation on
- even and odd functions.
- Now, let's see what sine of minus a is.
- Sine of minus a is equal to-- so this is minus a.
- So it's this side, so it's the minus length of-- let's call
- this x, let's call this y, and let's call this, well,
- let's leave that h, right?
- If that is x, this is y, this length is y, then this length
- right here is minus x, right?
- So the sine of minus a is minus x/h.
- What's the sine of a?
- Sine of a is equal to-- this is a-- opposite or hypotenuse,
- x over h, right?
- So sine of minus a is equal to minus 1 times x over h, right?
- Or this is just the same thing as-- I mean, we could multiply
- both sides of this by minus 1, minus x over h, right?
- So sine of minus a is equal to minus sine of a.
- So let me clear this out and rewrite this identity.
- And as you can see, all I'm doing is I'm just playing
- around with triangles and showing you that, you know,
- just using the basic SOHCAHTOA, you can actually discover a
- whole set of trigonometric identities.
- So let's clear that.
- And, you know, it is useful to memorize.
- I normally don't advocate memorizing, but it's helpful
- just to do things quickly.
- But I'd also advocate being able to prove it to yourself,
- so if you ever forget it, and you don't have a cheat sheet
- available, you can prove it, and if you ever have to teach
- it, then you'll be able to explain the underlying
- themes a little bit better.
- So let's clear this.
- Let's see if we can discover some more trig identities.
- So we know that-- so let's see, if we have sine--
- what's sine of a plus pi/2?
- a plus pi/2.
- Well, we could use our handy sine of a plus b identity,
- which we've already proved, so we can use it now.
- So that tells us that it's the sine of a-- that equals the
- sine of a times the cosine of pi/2 plus the sine of pi/2.
- And we're in radians, of course.
- This could have been 90 degrees instead, if we
- wanted to be in degrees.
- sine of pi/2 times the cosine of a, right?
- Well, this equals the sine of -- what's cosine sign of pi/2?
- Or cosine of 90 degrees?
- Well, that's when we're on the unit circle, we're
- pointing straight up.
- And so the x-coordinate is 0.
- I could draw it out, but I think-- you might want to draw
- the unit circle and figure it out for yourself, or if you
- don't, do it on a calculator, but you will learn
- that it is 0.
- The cosine of pi/2 is 0.
- Plus sine of pi/2, for the same reason, we're pointing straight
- up on the unit circle, so the y-coordinate, or the sine
- coordinate, is 1, right on the unit-- is essentially at the
- point 0, 1 on the unit circle.
- So sine of pi/2 is 1, and then times cosine of a.
- So sine of a times 0 is 0.
- 1 times cosine of a is just cosine of a.
- So we have a new, useful trig identity.
- Sine a plus pi/2 is equal to cosine of a.
- So really, this is just telling us that cosine of a is the same
- thing as sine of a shifted.
- So if we were to think of this graphically, if we were to
- think of, you know, if we were to draw the graph, if you shift
- the sine graph to the left by pi/2, you get the cosine graph.
- And if you haven't learned about shifting yet,
- don't worry about that.
- Or you might want to actually graph the two, and I
- think you'll get a sense of what I'm saying.
- So let's do-- I don't know.
- And another way to rewrite this exact same thing is the sine of
- a is equal to the cosine of a minus pi/2, right?
- Let's say I said that b is a plus pi/2, right?
- Let's say I said that b is equal to a plus pi/2, then we
- can say that this is b, and then this would b minus pi/2.
- I'm just switching around variables.
- I'm doing this in a much more loosey-goosey fashion than I
- normally do a lot of videos, but I want to show you that a
- lot of this trigonometry can just be-- you know, it's
- just kind of discovery.
- What's sine of a minus b?
- Well, that looks like a new one, doesn't it?
- Well, let's try to figure it out.
- Well, that equals sine of a cosine of minus b plus sine
- of minus b times the cosine of a, right?
- Well, what do we know about the cosine of minus b?
- Before I cleared the screen, we just figured out that the
- cosine of minus b, since it's an even function, is the same
- thing as the cosine of b.
- So we can rewrite that as that equals the
- sine of a cosine of b.
- And then what's the sine of minus b?
- Well, that's the same thing as the minus sine of b.
- That's what we just proved, that the sine of minus
- b, that this is equal to minus sine of b.
- You could draw the triangle and the unit circle, if you don't
- believe me, but we just did that.
- So we can say that that is equal to minus sine
- of b cosine of a.
- I encourage you do the same thing with the
- cosine of a minus b.
- These are all just, you know, we're using one or two or three
- trig identities together and trying to come up
- with new things.
- And I think at this point, we've literally gone over
- everything, that almost every trig identity you've seen in
- your book, you should be able to get there somehow,
- just by keep on playing.
- And obviously, all of these identities, you can invert the
- sines and the cosines and the tangents, and you can get
- identities for secant and cotangent and cosecant
- and keep playing around.
- And I encourage you to do so.
- And do it graphically.
- Draw the triangles.
- It's also interesting to sometimes actually draw the
- graph on the x-y plane of, say, you know, cosine of x plus
- pi/2, or sine of x plus pi/2, or sine of x.
- And I think in the future, I'll do a video where I really
- do explore all of that.
- Well, I hope I haven't thoroughly confused you.
- I wanted to just kind of show you that a lot of trig-- it
- all comes from SOHCAHTOA and playing around with SOHCAHTOA
- and triangles, and you can pretty much get-- you can
- pretty much solve for everything you learn
- in trigonometry.
- And if you don't have SOHCAHTOA, at least the unit
- circle definition, which is actually better, because
- it's more extensive.
- But anyway, that's all for now.
- See you soon.
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