Using Algebra Skills with Trig Expressions and Identities • [6.1a] PRE-CALCULUS 12

hey there in this video we are going to look at some of the algebra skills that you've hopefully learned in the past and connect with how you can use those same algebra skills in working with

simplifying expressions that involve trig functions so as you start working with trig identities you're going to need to take all of the algebra skills that you've

learned in the past and bring them forward now and apply them to working with new Expressions where the variables aren't just things like a and b the

variables are trig functions like sine X and cos x but fortunately all of the things that you've learned algebraic skills in the past are going to be applicable here you can work with them

exactly the same way it's just a question of learning to see that it's a similar thing so in this video what we're going to do is make the connection and have similar Expressions these are

actually fairly similar Expressions except that the variables look quite different so in the past you've learned that you can factor that expression it has a common factor of a so you could

write it with the a factored out and you would have a times 1 plus b now you could go back backwards and check to see that that works in other words you could multiply out to C you

could multiply this by this and this by this and you'd get a plus a b which is what we started with but the key is understanding that these two things are equivalent they're just

written in different forms and in the same way you can work with trig Expressions that way so here we have something with two terms that term in that term and it has

each term has that they have a common factor of sine X so just like the the expression over here we can write it in factored form we can factor out the sine

X so we can write it as sine X times 1 plus cos x and in just the same way you could multiply it out to show that it's the same work backwards again to check your work

sine X Plus sine X cos x right because sine X times cos x is sine X cos x there so again though

those two expressions that one and that one are equivalent all right there's another one down below here that we're going to look at this expression is one that can be factored now it doesn't have a common factor but in fact this is

called a difference of squares it's something squared minus something squared so it can be written as two binomials that are very much the same

this first term is M squared and I'm hoping you've seen this before this idea that those two binomials are both going to have M as

the the leading term in them and that's a nine so they're both going to have a 3 as the the second term but the key difference here is one of them is going to be plus and one of them is going to

be minus in between now you can check to see that that multiplies out you can work backwards again you can multiply M times M gives you the M squared and then

you can multiply the outside terms you get minus 3M and you can multiply the inside term and you get plus 3M I'll write it down there and then you can multiply the last terms and you get

minus nine so it's m squared these add to zero so they're not even represented in our final expression here of that all right so again what that is

is this expression is equivalent to that expression right one's in factored form one's in expanded and simplified form

now in just the same way this is a difference of squares because it is something squared one one is one squared and sine squared is sine squared and the

name of what we call it now remember that this squared here actually you could write this as 1 minus if you wanted to you could think about it it's the sine ratio squared so that oh that

variable whatever the value of sine X is it's that number squared but in just the same way you could write this as two binomials like that one one

of them is going to be one plus sine X one of them is going to be one minus sine X all right so this expression can be written in factored form as that

expression again you can double check by working backwards one times one gives us the one the outside terms

minus 1 sine X plus one sine X those would add to zero and then that gives us the minus sine squared there all right let's have a look at a couple of other

algebraic expressions and see what we can do with them so we have a couple of fractions here that we're going to add together and they have different denominators so if they have different denominators you're

going to need to turn them into expression to where each of them has a same denominator common denominator and that common denominator if this one has an A and this one has a b that common

denominator is going to have to be a b a times B so to do this what we're going to have to do then is we're going to have to multiply this one by

B over B and this one by a over a right if we multiply this fraction by B over b b over B is actually equal to one

anything divided by itself is one so when you multiply by that you're not changing the value it's going to be in a different form right we're going to have

1 times B we're going to B over a b but the value of this is the value same as the value of what we started with there same goes for the second fraction one

over B times a over a is a over a b and now that they have a common denominator we can add them together we can write it as B plus a or maybe I'll

write it as a plus b if you wanted in alphabetical order there it doesn't matter when you're adding like that over that single denominator when you add fractions the denominator is what it is

a b is the denominator of the two separate fractions or of this combined fraction like that and let's remember again that original expression is the same as this

says simplify up here it might not look simpler to you it looks like there's actually more to it but this the advantage here is that this is a single fraction which is useful sometimes

instead of two separate fractions now we can do the exactly the same thing with this expression over here it's got trig functions instead of just variables

but we can do exactly the same thing maybe I'll rewrite this one so that we can write our show our work a little better here

so we have that and just the same way here we have to make a common denominator now the common denominator is going to be these two things multiplied together

sine X cos x now this is rated up there to remind us so we're going to have to multiply this first one by sine X over sine X

and the second one by cos x over cos x and so what we end up with there is we have

sine X over sine X cos x and we have on this one cosecs on top cos x times one over sine X cos x

and then once you've created two fractions that have the same denominator you can add them together so on the top since those are different sine X Plus cos x I just have to write it like that

just like I did over here those are different variables they're not like terms so I can't combine them together but then I have this single denominator on the bottom of sine X cos

x and just as for the other one this is now a single fraction which is often more useful when you're working with trig identities then these two separate fractions even though this actually may

look a little bit simpler to you all right the second one down here is very much a similar thing although you have a fraction and then something that does not look like a fraction but in much the

same way you can combine those together to make a single fraction out of it and I'm hoping you've seen in the past here that if you're trying to combine something that's a fraction with

something that's not a fraction you can imagine the second thing is a fraction as one and what we need to do here is we have to give it the same denominator so since

the only thing that in our expression that has a denominator is this one the denominator is a we can use a as the common denominator and to get that we're going to multiply this by a over a if it

helps you you can think about that one there and what you end up with then is 1 over a minus a squared over a and

then we can do what we did up above which is once they have the same denominator we can write it as a single fraction one over a squared 1 minus a squared over a and again that

is sometimes a simpler expression to work with than that original one and we can do the same thing here if we want to create this turn this into a a

single fraction expression we can multiply the second thing by cos x over cos x because that's our denominator the only thing that has a denominator and is that first one so as long as we turn the

second one into something that has that same denominator I will write it down below here 1 over cos x minus cos

squared x cosine times cosine there is cos squared and over cos x and then we can write it as a single

fraction one minus cos squared over cosine X all right so again that expression is equivalent to that we've just changed the form all right we'll look at a couple of

other things here that might be useful as you work with trig identities again some more work with some fractions here and just want to help you recall something that you've undoubtedly seen

before but maybe not in this form that you often work with with trig identities which is uh something that are called complex fractions so this is a fraction

that itself has in the numerator in the denominator two fractions so it's like fractions within fractions that's what we call a complex fraction

now in this particular case we have fraction over another fraction but we can think about it as a division because this line here is how we write division

that's the nature of what a fraction is so we can think of it as this fraction divided by this fraction now probably when you first learned how to divide fractions you saw it looking like this

with the division symbol like this over that right A over B divided by a over D and you probably learned that dividing

by a fraction it's the same as multiplying by the reciprocal of that second one of

the one you're dividing by so we can change this into A over B times D over a we can flip that

second one over you can flip this one over to that and then you can sometimes then look to simplify things so in this case actually here my

a didn't get written very well so I'm going to fix that that's actually supposed to be an A on the bottom because I flip that over pretty key in this situation here I want to have my D's and A's looking

the same but since we have on our expression here we have an A in one of the tops of the fractions and an A in one of the bottoms and we're multiplying if we're multiplying there we can divide

those out in other words we often just call it canceling them out right if you divide the top and the Bottom by a they're going to disappear there and you're going to just have D on the top

and B on the bottom so this expression is the same as this expression in a much simpler form which is often useful for what you're doing and in exactly the same way here we can work with this

complex fraction now it's important to keep track of where the kind of the main division bar is there so if we wanted to we can say that this

is the same as sine X over cos x divided by sine X over secant x or we can even just go straight to the

next step here and say if I have this thing divided by this it's the same as that first thing if I know I'm dividing by this I'm dividing by that I can just change it

right away and go straight to this and change it to multiplying by the reciprocal of this which is secant x over sine X and then you can look to do the same

thing that we did over here if it has something in common on the top and the bottom and it does we're multiplying we've got a sign there and a sign there so we can divide those out cancel them

and then we're going to end up with secant x over cos x so this is a simpler version than this now actually later in your work with

trigger identities you're going to see that you could actually simplify that even more but we're going to leave that for now all right just a couple of other things here that as you're working with trigonities you're going to have to think about some of the things that

you've learned or learned not to do and so just want to point this out here as an example of of that are each of these expressions

correctly simplified so if you have a plus b over C can you turn that into a over C plus b over C well in fact you

can because that's exactly what we just did a few examples ago where we were adding fractions we were taking two fractions that had a common denominator and putting them together like this we're going that way but you can

certainly go the other way if you have something that has more than one term on top but a single term on the bottom you can split it up and you can divide each one separately you can write it as a

divided by C plus b divided by C all right but what you can't do is this one can't split the denominator if the denominator is the thing that has two

things added together you can't split it up in that way you can't say it's C divided by AC divided by B so this one is a definite yes that's okay this is a definite no if you want to see that that

this is a definite note just just put some numbers in there to to see to try it out like let's say we've got uh let's say C was five and the bottom a was two

and B was three so five over two plus three is the same as five over five which is one right so we know that in my case here with the variables I picked

that this side equals one so if this were something that you could do to simplify then certainly if I put the numbers in on this side

it would work right so if I put my 5 in there for C and I put my 2 in for a and I put my 5 in for C here and I put my 3

in for B you can see already that this isn't going to work because this is way more than one this is way more than one it doesn't even matter specifically what it is but it's definitely more than this

when you add it together all right so that is Def that's a definite no that doesn't work all right so there's lots more algebra skills that you've undoubtedly learned in the past

and this is just a sampling of them to give you the idea of how you can start to transfer those algebra skills to working with Expressions that involve trig functions and work with trig

identities all right that's it [Music]