Trig identities part 2 (part 4 if you watch the proofs) More playing around with trig identities
Trig identities part 2 (part 4 if you watch the proofs)
- Welcome back.
- I'm now going to do a bit of a review of everything we've
- learned so far about maybe even trigonometry and
- trig identities.
- And then we'll see if we can come up with-- maybe use what
- we already know to come up with a couple more trig identities.
- So we know that from SOH-CAH-TOA we know that sine
- of theta is equal to the opposite over the hypotenuse.
- Let me draw a triangle here.
- If I were to draw a triangle here-- whoops.
- Oh, there you go.
- OK, so this is theta.
- This is the opposite.
- This is the adjacent.
- This is hypotenuse.
- Then sine of theta is equal to opposite over hypotenuse.
- Cosine of theta-- this is basic review, hopefully at this
- point-- is the adjacent over the hypotenuse.
- The tangent of theta is equal to the opposite over the
- adjacent, which is also equal to the sine of theta over
- the cosine of theta.
- And we showed this in a couple of videos ago.
- And then, these are kind of almost definitional, but the
- cosecant of theta is equal to the hypotenuse over the
- opposite, which is the same thing as 1 over sine of theta.
- You can just memorize this.
- I mean, I kind of find is silly that there is such
- a thing as cosecant.
- I guess it's just for convenience because you know
- everyone knows it's just 1 over sine of theta.
- And same thing for secant.
- Secant of theta-- it's really for convenience.
- Instead of having to say, in the case of secant-- oh, that's
- 1 you know-- if you end up with the equation 1 over cosine of
- theta you can just say, oh, that's just the
- secant of theta.
- I think it actually has some obvious properties and if you
- were to draw unit circle and all of that too.
- But anyway, so that's equal to the hypotenuse over the
- adjacent, which is equal to 1 over cosine of theta.
- And then, of course, cotangent of theta is equal to the
- adjacent over the opposite, which is equal to
- 1 over tan theta.
- And of course, that's also equal to cosine of
- theta over sine theta.
- It's just the opposite of the tangent of theta.
- Or that's the same thing as what?
- That's the same thing as the secant-- no, no, no.
- It's the same thing as the cosecant-- no, no no.
- Let me make sure I get this right.
- It's the same thing as the-- I just want to get the inverses.
- Well, let's prove what it is actually.
- I always confuse myself.
- So this is the same thing as 1 over the secant of theta over
- 1 over the cosecant of theta.
- Secant of theta, cosecant theta.
- And then that equals the cosecant of theta over
- the secant of theta.
- I wouldn't waste your time memorizing.
- So we know that a cotangent of theta is equal to
- 1 over tangent theta.
- Is equal to the cosine over the sine.
- And it also equals the cosecant over the secant.
- And I wouldn't worry about really memorizing this.
- You could derive it if you had to.
- As you could tell, I really didn't have
- this memorized either.
- And we also learned in previous videos that the sine squared
- theta plus the cosine squared of theta is equal to 1.
- And that just comes from the pythagorean theorem.
- And if you play around with this a little bit you'd also
- get that the tangent squared theta plus 1 is equal to
- the secant squared theta.
- You actually go from here to here if you just divide both
- sides of this equation by cosine squared.
- So we know that.
- And then if you've watched the last two proof videos I made,
- we also know that the sine of-- let's say a plus b-- is equal
- to the sine of a times the cosine of b.
- Plus-- let me erase some of this because I don't
- think that that is an important trig identity.
- You can derive it on your own.
- I just wanted to show you that you could figure it out.
- I'm using too much space.
- OK, now I have space.
- Let me find that blue color I was using and make
- sure my pen is small.
- So it's the sine of a times the cosine of b plus the sine
- of b times the cosine of a.
- And you might want to just memorize it.
- This actually becomes really useful when you actually start
- doing calculus because you have to solve derivatives and
- integrals that you might have to know the identity.
- And it's not that hard to memorize.
- It's the sine of one of them times the cosine of one of them
- plus the other way around.
- That's all this is.
- And then we also learned that the cosine of a plus b-- it's
- the cosine of both of them minus the sine of both them.
- So that is equal to the cosine of a times the cosine of b.
- And I proved this in another video, hopefully did it
- to your satisfaction.
- Minus the sine of a times the sine of b.
- These are pretty useful because from these can we can come up
- with a bunch of other trig identities.
- For example, what is the sine of 2a?
- Well, that's just the same thing as the sine of a plus a.
- And if we use this trig identity up here, that is equal
- to sine of a cosine of a plus the sine of a, cosine of a.
- I just used this sine of a plus b identity up here and well,
- want a and b are both a.
- Now what does this equal?
- Well, this is two terms that are just both sine
- of a, cosine of a.
- So that just equals 2 sine of a, cosine of a.
- So we now have derived another trigonometric identity that
- might be in the inside cover of your trig, or actually,
- your calculus book.
- All of these actually, I could draw a square
- around all of them.
- Let's do another one.
- Once you have a bit of a library of trig identities you
- can really just keep playing around and seeing what else you
- can-- and I encourage you to do so.
- And you'd be amazed how many other trig identities
- you could come up with.
- For example, let's do cosine of 2a.
- Cosine of 2a is equal to cosine of a plus a.
- And cosine of a plus a, what did we say?
- It's the cosine of both of the terms times each other minus
- the sine of both of the terms.
- So that equals cosine of a, cosine of a, right?
- Cosine of a times cosine of a minus sine of a, sine of a.
- This identity was the cosine of a plus b identity.
- Minus sine of a.
- So what is this?
- This is equal to cosine squared a minus sine squared a.
- That's interesting.
- We could play around.
- This is interesting because this is the form a
- squared minus b squared.
- So that's also the same thing as a plus b times a minus b.
- So that's the same thing as cosine of a plus sine of a
- times cosine of a minus sine of a.
- I don't know.
- This isn't really a trig identity, but I'm just showing
- you could play with things.
- Cosine of 2a is equal to cosine of a plus sine of a times
- cosine of a minus sine of a.
- So the sum of the cosine and sine of a then
- times the difference.
- That's just interesting.
- I'm just showing you that what's fun about trigonometry
- is you can kind of keep playing around with it.
- And actually, that's probably-- that is how all of the trig
- identities were discovered.
- So let's say that we have-- we want to figure out what cosine
- of let's say, negative a is.
- Well, let me draw a right triangle.
- That's almost a right triangle.
- Now let's say this angle is a.
- So negative a, unit circle would look something
- like this, right?
- Negative a.
- So cosine of a, if we say that this side is the adjacent
- side, this is the hypotenuse.
- This would still be the hypotenuse, right?
- And this is the opposite.
- This is the negative opposite.
- So cosine of minus a is equal to what?
- This is minus a, so it's the adjacent over the hypotenuse.
- So it equals the adjacent over the hypotenuse,
- which we just say is h.
- But that's the same thing as cosine of a, right?
- Because cosine of a is also the adjacent over the hypotenuse.
- Oh, I'm almost out of time.
- Let me switch to a new video.
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