hey there in this video we are going to look at how you can determine and express restrictions on trig identities and expressions so trig expressions and identities can have restrictions or in other words non-permissible values are what you might have called in the past in PVS for short and those are on the variables that are involved just like other expressions and equations that you've seen and worked with in the past but for trig identities these might happen for several reasons first of all the Expressions involved might have certain limitations for algebraic reasons just like the Expressions that you've worked with in the past often having to do with the denominator or a square root involved or something like that but the other reason is that the trig functions themselves might have certain values that are not permissible as you've likely already seen in this unit so we have a couple of examples we're going to look at here so this first one this is an expression and we're going to work out the restrictions on that expression so first of all of my two things in my list up here the Expressions involved if we look at those we have sine X and cos x now those two things themselves there's no restrictions on sine and cosine they're defined for all values of X so no restrictions arise because of that however for this second Point here we're going to have a restriction on this expression it has to do with the denominator here when you have an algebraic expression that has a denominator with a variable in it you can have some restrictions that are a result of that essentially in this case that cos x minus 1 the denominator cannot be zero so we're going to have restrictions because of that that the denominator cannot be zero so we have cos x minus 1 cannot be zero or in other words if we go about solving that inequality we have cos x cannot be one and then we have to think about what values of X are going to satisfy this all right and then we have to think about what values of X we're looking at there I think the simplest way to think about what values of X you're thinking about where is cosine one I like to think about the graph when I try to think about what values of X that is so just think about the graph of cosine cosine starts at 1 up there it hits 1 again at 2 pi it hits 1 again at 4 Pi it would hit 1 the other way at negative 2 pi and so on right so if you made a list of those I forgot 0 in there but the way you can express that is you can just say x cannot be 2 pi times n where n is some integer and is any integer so you can express the restrictions certainly like this but sometimes you also see the restrictions written like this in terms of another trig function or specifically in terms of sine and cosine all right so that's that one we're going to look at a second one here so in this situation this is an identity right there's two sides to it it's this equation here that's an identity we're going to determine the restrictions on it we're still going to look at the two different aspects involved here so first of all we're going to look at the trig functions involved so those trig functions involved there first of all the sine and the cosine themselves have no restrictions so we'll just note that again here but the other ones involved here namely secant and tangent themselves have some restrictions so tangent tangent is equal to sine Theta over cos Theta so if you think about what it is in terms of sine Theta and cos Theta you can make sense of what restrictions it has tangent is undefined where cosine that denominator is zero so the restriction on tangent is cos Theta cannot be zero and if you think about it again I would go with the graph here think about the graph cosine where it's zero cosine starts at one it's zero there it's zero there there's two Pi again and it's zero there it's zero there and there's four Pi so basically we have at pi over two three pi over two five pi over two seven pi over two if you continued it the other way you'd have negative pi over two and so on so the values involved in that are Theta can't be pi over two plus multiples of Pi so we're going to write that as pi over two plus pi n right now the other thing is secant secant Theta or second function involved that has restrictions secant is equal to one over cos Theta so for the same reasons as above cos Theta can't be zero and Theta can't be those values so we don't need to write that again but it's exactly the same values there now the second thing is here for the Expressions involved here so if you think about the Expressions involved here this is going to have to do with the denominators here so in this case this denominator cannot be zero and this denominator cannot be zero so we'll look at those two one at a time here so what we have here is that one plus tangent can't be zero or in other words tangent can't be negative one if you isolate that tangent on there for this it's not necessarily if you're familiar with the graph of tangent you can think about the graph of tangent the way we did for cosine but thinking about where tangent is negative one or maybe what's simpler is think about where tangent is one tangent is one where the angle the reference angle is pi over 4 45 degrees pi over 4 because you think about that little triangle there like that and where it's negative one specifically is going to be in quadrant two and four so if that was pi over four and that was pi over 4 you'd have 3 pi over 4 right as the first one you'd have 7 pi over four if you went around again here you would have 11 pi over four and so on the values that it can't be there so if we're going to try and write an expression for what those values can't be we're thinking it's 3 pi over 4 plus half turns plus multiples of Pi half turns are Pi so we're going to pick the first one that one and add or subtract multiples of Pi here so it's 3 pi over 4 plus pi times n where n is an integer and then we'll look at the other fraction on the other side of that identity so the other denominator is sine squared and we know that can't be zero sine squared being zero if we square root both sides the simpler version of that is sine can't be zero and if we think about where that is this is a case where I would think about the graph because the sine and cosine graph are not too difficult to think about what they look like so there's the sine graph more or less and we're thinking about where it's zero so it's zero at zero at Pi at two Pi at 3 Pi at negative pi and so on so if we're trying to write an expression for that what we're going to write is we're going to write that Theta cannot be this is multiples of Pi here right multiples of Pi so we're going to say pi times n and that's what we have now so we've got all of our restrictions sorted out there so we're going to put them all together so essentially we want to do here is we want to take the different ones that we have so we have this one from the tangent and the secant we have this one from the 1 plus tangent in the denominator and we have this one from the sine squared in the denominator so if we put all that together we're more or less saying that Theta cannot be pi over 2 plus pi n Pi n and then 3 pi over 4 plus pi n now we can actually simplify this it's good to write it in the simplest form possible and the reason you can simplify this is if you think about what values each of these represents what that represents and what that represents so let's start with this one this represented 0 by 2 pi 3 pi and so on negative pi but this one represented starting there at pi over 2 pi over 2 3 pi over 2 5 pi over two negative pi over 2 it fills in all the ones in between if you think about it going around an angle in standard position you have zero pi over two pi 3 pi over two pi and so on it's every quarter turn like that so the simplest way to write that is just to say it is multiples of pi over 2. all right it's multiples of pi over two it's multiples of a quarter turn so those two together can be combined and we can just say in simplest form this is pi over 2 n and then of course we need to include this other one here which is 3 pi over 4 plus pi n so that's our restrictions in simplest form for that identity all right so that's a look at how you can determine and express restrictions on trig identities and expressions [Music] thank you [Music]