Pre-Calculus: Factoring Trigonometric Expressions

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Factoring Trigonometric Expressions
Now another thing we can do with trig expressions is sometimes factor it just like we factor regular kind of expressions. And, again, we can look for common factors or we could try to foil, untangle and foil and so forth, and factoring in some standard ways. Let me sort of illustrate how one could do this with trig functions, remember that sometimes simplifying first can save a lot of effort later.
So, how about this one? Sin? + sin? x tan?. Well, let’s see if we can simplify this by factoring. Well, I see that I can factor in a common factor of sin?. So this equals sin? x, and here I’m left with an invisible 1, not zero, because when I distribute back, I’ve got to get that term there, so I need a 1 there to get me going. And then, plus, and this can be replaced by just tan?. But 1 + tan?, remember that’s one of our consequences of the Pythagorean Theorem. 1 + tan? just equals secant?. So, in fact, I can replace this by sin? times sec?. And, in fact, we could simplify this a little bit more, I guess, because what is secant ?? Secant ? is 222222222222221cos. So this equals 2sin?times 21cos? and so that’s just 2sincos???????. But sincos1cos, that has a name. We call that tangent. So, that’s tangent2?. So, again, just factor out the sin and I’m left with 1 + tan. 1+ tan is secant, by the Pythagorean Identity. Secant is 22221cos, so I see , which is tan. So, in fact, this can be simplified really quite nicely, down just to this. The power of factoring together with some of these identities. 2cos2
Let’s try this one together. How about sin(x) - cosx - 1. Doesn’t look like we can factor this at all, because I’ve got sines and I’ve got cosines. How do you factor those? Well, you might not be able to. However, what I see is sin + cos = 1. So I should be able to use that identity to convert the cos to a sin. So, let’s see if we can do that. So, let’s come over here and do a little mini-calculation. I remember that sin x + cos x = 1, then what does cosx equal? It would equal 1 - sinx. so, in place of this cos x, I’m going to put in 1 - sinx. Okay, so this equals the first sin(x) -, and then instead of cosx, I’m going to put in 1- sinx, and then I have a -1 there. Does that look good to you? I hope it doesn’t look good to you, because I’ve made a Classic Mistake. Do you see this Classic Mistake I made? Well, it’s Classic Mistake Number Four. Number Four -- the subtracting mistake. Because, remember, if you’re going to subtract this cosx, you’ve got to subtract all of this stuff. All I did was subtract the first term, I never bothered to subtract this term. So, in fact, major Classic Mistake here and if you were fooled, that’s great, because then you should realize that that’s something to watch out for. That negative sign has to, in fact, distribute everywhere. Remember, “share the negativity.” 22222222222222
Well, now what do we have? This equals sin x- 1, and then distributing that minus sine, it becomes a plus sinx, and then I have that -1 still out there. And now what does that equal? Well, now I can rewrite this a little bit. I’m going to write this term first. I’m going to write the sinx first, that’s just this term. And then I’ll write this term plus sin(x) and then I have a -2. And now the question is, can somehow, this be factored? Well, let’s try. Now, usually we factor things like something squared plus something minus two. Now, it may be easier for you just to do that first. It may be easier for you just to consider something squared plus something minus two, and try to factor that. If it’s easier to do that, then that’s fine. They have opposite sines and let’s see, I think I should put the 2 here and the 1 here, and that actually works out perfectly. So, you could do that and then remember that this is actually sin(x), or the other possibility is just to try to factor this immediately, put a sin(x) and a sin(x) here. And a plus and a minus, and then a 2 and a1. And now, it’s factored. So, in fact, this expression, even though it doesn’t look like it can be factored at all, after you put in some trig type identities, can be factored quite tidily into this. 22
Let’s try one last one. 22sincxxcsx++. All right, now what do you do. Well, again, I don’t see any kind of common thing here, until I remember what cosecant is. Cosecant is actually 1sin. So, it’s in the denominator, so I see 1sin in
the denominator. When I invert and multiply, I see sin1. So, in fact, this really is the same thing as 2sin2sin1xx+. Because, remember 1csc is just 1. So, whoever gave us this problem was really sort of demonic and mean because they were trying to camouflage what otherwise would be a very straightforward, simple looking thing.
Now, this maybe we can factor. And let’s see. I’ll put a sin(x) here and a sin(x) here and I see the signs are going to be the same, they’re both plus and how about 1 and 1? Well, that works. Sin. Sine plus sine is 2sin(x), 1+1 is 1. So, in fact this can be factored quite nicely. It’s a perfect square in fact. (Sin(x) + 1). So this awful looking thing right here turns out to be, in factored form, just (sin(x)+1). Look at the power of first using the trig identities to rewrite complicated looking things, the things that look a lot more smoother, and in factoring the smoothness and get something really nice. The power of using trig together with factoring. Pretty powerful stuff. Try these and see if you have some fun. 222