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Trig limit using double angle identity

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.E (LO)
,
LIM‑1.E.1 (EK)
Sal finds the limit at θ=-π/4 of (1+√2sinθ)/(cos2θ) by rewriting the expression using the cosine double angle identity.

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  • leafers tree style avatar for user Math Mind
    Wow, my Trig course never taught me fully the formula at , at least I only have part of it in my notes. I've never seen the =1-2sinx^2=2cosx^2-1 before.
    Can anyone direct me to which video he discusses that? And is there a similar formula for Sin2x? I am wondering if there is a portion of that I never knew as well. Thanks :)
    (58 votes)
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  • blobby green style avatar for user Adithya K Challa
    At how did Sal come up with the open interval at (-1,1)? What is the logic behind the open interval?
    (24 votes)
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  • spunky sam blue style avatar for user luke.padgett.1
    At , how did Sal know off the top of his head what sin((-pi/4)) was? Is there a series of videos I could watch to learn these radian values?
    (9 votes)
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    • duskpin sapling style avatar for user Vu
      -π/4 is an angle on the unit circle. Most of us have it memorized. You may want to memorize them as well for future benefits if you continue to do math. You can google "unit circle" and get a lot of pic/charts showing those special angles and their cosine's and sine's values.
      (19 votes)
  • mr pink orange style avatar for user Steve L
    At in the video, Sal begins to explain that it is important to understand that the factored equation is not the same as the original unfactored equation. In his explanation he speaks of two functions f(x) and g(x). I’m assuming that g(x) is the original unfactored equation and that f(x) is the resulting factored equation. In the video, Sal states that f(x) and g(x) are equivalent except at ‘a’, and that f(x) is continuous at ‘a’. I'm assuming this means that g(x) is defined for all real numbers except 'a', and therefore is not continuous at 'a'. And that f(x) is defined and is continuous at ‘a’. Sal goes on to say that the limits of f(x) and g(x) are equivalent at 'a'.

    Is there a mathematical theorem or proof that states the limit of a factored equation is equivalent to the limit of the respective unfactored equation as long as the factored equation is continuous at the point for which the limit is taken? Is this distinction covered by the Epsilon-Delta proof of limits. For I learned from the video on Epsilon-Delta proofs that a limit can still exists at 'a' despite a function being undefined at 'a'.

    I want to better understand this distinction as Sal has identified this situation several times in multiple videos, i.e. where a factored equation has removed a discontinuity which was present in the original unfactored equation, and that the limits are considered equivalent. I believe mathematicians refer to such a discontinuity as a ‘removable discontinuity’.

    I am comfortable performing such manipulations, as it makes inherent sense to me that the limits are equivalent; but I wouldn’t know how to mathematically state or prove that such manipulations are correct. If a function can be factored, and the factoring removes a discontinuity, it seems more logical to me to state that the original function had a false or phantom discontinuity. In other words, the discontinuity doesn’t really exist, it only seems to exist due to the equation not being in its simplest, i.e. factored, form.
    (14 votes)
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    • starky sapling style avatar for user tejas_gondalia
      I have already given a same answer to a similar question, but still I'll hope this helps you out.
      As Sal has mentioned in his introductory videos that "The beauty of limits is:They don't depend on the actual value of the function at that limit. They describes how the function behaves when it gets close to the limit."
      So here in this case, Sal created an analogy by creating the refactored function f(x) which is continuous at x=a and then as g(x) is same as f(x) except it is discontionuous at a, so from the definition stated above, limit of g(x) has to be equal to f(x) at that point.
      And the way we can evaluate the limit of f(x) at x=a is by various methods like epsilon delta, graphs, tables etc. which are logically as well as mathematically correct.
      So, long story short, Sal evaluated the limit of the original function the other way around(which is correct :-)).
      I'm sorry if i was unclear. Just a small effort to help. :-)
      (6 votes)
  • blobby green style avatar for user Michael Beans
    how does 1-2sin^2(theta) get factored into squrt(2)sin(theta)? that doesn't make sense to me
    (8 votes)
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    • blobby green style avatar for user be.cse.deepak
      we are seeing 1-2sin^2(theta) in the form of a^2-b^2 which is (a+b)(a-b) and we can even write 2 as ( sqrt(2) )^2 and 1 as 1^2 , since square and square root will cancel each other and 1^2 is 1.
      now we can write 1-2sin^2(theta) = 1^2-( sqrt(2)sin(theta) )^2= (1+sqrt(2)sin(theta))(1-sqrt(2)sin(theta).
      and after that we can cancel out the common factor from numerator and denominator.
      (9 votes)
  • blobby green style avatar for user Nawfal Baghyour
    at how is cos2(theta)= cos^2(theta)-sin^2(theta) what is the difference between cos2 and cos^2 ? thank you
    (6 votes)
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  • starky ultimate style avatar for user Jeff Pettigrew
    at you have conver 2 cos pi/4 as the same as cos pi/2...
    my question is why do must you convert in this way as the cos pi/4 is sq root 2 over 2... x 2 is the sq rt of 2.. and the cos of the sq rt of 2 is not 0
    (3 votes)
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    • duskpin sapling style avatar for user Vu
      It is cos(2θ) which is not the same as 2cos(θ). Sal should had use parenthesis like I did to avoid confusion for students. cos(2•π/4) , you simplify what inside the parenthesis first then compute cosine function.
      (8 votes)
  • male robot hal style avatar for user Arush Bansal
    what is sal referring to in open interval at .
    (3 votes)
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    • leaf green style avatar for user kubleeka
      The interval (a,b) is just the set of real numbers between a and b. If we mean to include the endpoints a and b in the set, we write it as [a,b] and call it a closed interval. If we mean to exclude them, we write (a,b) and call it an open interval.
      (5 votes)
  • leaf green style avatar for user akshajarora1
    In the previous problem, it was explained that lim f(theta) as theta tends to 0 = lim g( theta) as theta tends to 0(and f(0) was not defined)........where as in the end of this video it is written that : f and g are equal for x not equal to a and f is continuous at a THEN limit f(x) as x tends to a = limit g(x) as x tends to a...this seems a bit contradictory ...can anyone help me with this ?
    (4 votes)
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    • starky sapling style avatar for user tejas_gondalia
      Because Sal has mentioned in his introductory videos that "The beauty of limits is:They don't depend on the actual value of the function at that limit. They describes how the function behaves when it gets close to the limit."
      So here in this case, Sal created an analogy by creating the refactored function f(x) which is continuous at x=a and then as g(x) is same as f(x) except it is discontionuous at a, so from the definition stated above, limit of g(x) has to be equal to f(x) at that point.
      (1 vote)
  • starky ultimate style avatar for user DrewMTR
    At , Sal talks about something called open interval. Can someone explain it to me?
    By the way, is the initial function also 0/0 at 7π/4 and 15π/4 etc.? If so, then shouldn't the constraint be θ ≠ -π/4 + 2πk, where k is an integer? are are we not being this specific here?
    (2 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      An open interval doesn't include the endpoints.

      Example:
      The open interval 𝑥 ∈ (−2, 3) ⇔ −2 < 𝑥 < 3
      The closed interval 𝑥 ∈ [−2, 3] ⇔ −2 ≤ 𝑥 ≤ 3

      I don't know why Sal keeps reminding us that we're looking at this function of ours over an open interval, because it could just as well be a closed interval.

      – – –

      The function 𝑓(𝜃) = (1 + √2 sin 𝜃)∕cos 2𝜃 is defined for all 𝜃 such that
      cos 2𝜃 ≠ 0 ⇔ 2𝜃 − 𝜋𝑘 ≠ arccos 0 = 𝜋∕2 ⇔
      ⇔ 𝜃 ≠ (2𝑘 + 1)𝜋∕4, 𝑘 ∈ ℤ

      However, by restricting the domain to 𝜃 ∈ (−1, 1) Sal tried to make all discontinuities redundant apart from 𝜃 = −𝜋∕4, and had he instead chosen something like 𝜃 ∈ (−1, 0) he would have succeeded, because there's no need to list 𝜃-values for which 𝑓(𝜃) is undefined if they're not in the domain.
      (4 votes)

Video transcript

- [Voiceover] All right, let's see if we can find the limit of 1 over the square root of 2 sine of theta over cosine of 2 theta, as theta approaches negative pi over 4. And like always, try to give it a shot before we go through it together. Well, one take on it is well, let's just, let's just say that this is going to be the same thing as the limit, as theta approaches negative pi over 4 of 1 plus square root of 2 sine theta over the limit as theta approaches negative pi over 4. Make sure we can see that negative there, of cosine of 2 theta, and both of these expressions are, if these were function definitions or if we were to graph y equals 1 plus square root of sine, square root of 2 times sine theta, or y equals cosine of 2 theta, we would get continuous functions, especially at theta is equal to negative pi over 4, so we could just substitute in. We'll see well this is going to be equal to this expression evaluated at negative pi over 4, so 1 plus square root of 2 times sine of negative pi over 4, over cosine of 2 times negative pi over 4. Now, negative pi over 4, sine of negative pi over 4 is going to be negative square root of 2 over 2, so this is negative square root of 2 over 2, we're assuming this is in radians, if we're thinking in degrees, this would be a negative 45-degree angle, so this is one of the, one of the trig values that it's good to know and so if you have, if you have 1, so let's see, actually, let me just rewrite it, so this is going to be equal to 1 plus square root of 2 times that is going to be negative 2 over 2, so this is going to be minus 1, that's the numerator over here. All of this stuff simplifies to negative 1 over, this is going to be cosine of negative pi over 2, right? This is negative pi over 2, cosine of negative pi over 2, if you thought in degrees, that's going to be negative 90 degrees. Well, cosine of that is just going to be zero, so what we end up with is equal to zero over zero, and as we've talked about before, if we had something non-zero divided by zero, we'd say, okay, that's undefined. We might as well give up, but we have this indeterminate form, it does not mean the limit does not exist. It's usually a clue that we should use some tools in our toolkit, one of which is to do some manipulation here to get an expression that maybe is defined at theta is equal to, or does not, is not an indeterminate form, that theta is equal to pi over 4 and we'll see other tools in our toolkit in the future. So let me algebraically manipulate this a little bit. So if I have 1 plus the square root of 2, sine theta, over cosine 2 theta, as you can imagine, the things that might be useful here are our trig identities and in particular, cosine of 2 theta seems interesting. Let me write some trig identities involving cosine of 2 theta. I'll write it over here. So we know that cosine of 2 theta is equal to cosine squared of theta minus sine squared of theta which is equal to 1 minus 2 sine squared of theta which is equal to 2 cosine squared theta minus 1, and you can go from this one to this one to this one just using the Pythagorean identity. We proved that in earlier videos in trigonometry on Khan Academy. Now, do any of these look useful? Well, all of these three are going to be differences of squares, so we can factor them in interesting ways, and remember, our goal at the end of the day is maybe cancel things out that are making us get this zero over zero, and if I could factor this into something that involved a 1 plus square root of 2 sine theta, then I'm going to be in business, and it looks like, it looks like this right over here, that can be factored as 1 plus square root of 2 sine theta times 1 minus square root of 2 sine theta, so let me use this. Cosine of 2 theta is the same thing, cosine of 2 theta is the same thing as 1 minus 2 sine squared theta, which is just a difference of squares. We can rewrite that as, this is a-squared minus b-squared, this is a plus b times a minus b, so I can just replace this with 1 plus square root of 2 sine theta times 1 minus square root of 2 sine theta, and now, we have some nice cancelling, or potential cancelling that can occur, so we could say that cancels with that and we could say that that is going to be equal, and let me do this in a new color, this is going to be equal to, in the numerator we just have 1, in the denominator we just are left with 1 minus square root of 2 sine theta, and if we want these expressions to truly be equal, we would have to have them to have the same, if you view them as function definitions, as having the same domain, so this one right over here, this one we already saw is not defined at theta is equal to negative pi over 4, and so this one, in order for these to be equivalent, we have to say that this one is also not, and actually, other places, but let's just, let's just say theta does not, does not equal negative, negative pi over 4, and we could think about all of this happening in some type of an open interval around negative pi over 4 if we wanted to get very precise, but if we wanted to, for this particular case, well, let's just say, everything we're doing is in the open interval, so in, in open interval, in open interval between theta, or, say, negative 1 and 1, and I think that covers it because if we have pi, if we have pi over 4 that is not going to get us the zero over zero form, and pi over 4 would make this denominator equal to zero but it also makes, let's see, pi over 4 also will make this denominator equal to zero, 'cause we would get 1 minus 1, so I think, I think we're good if we're just assuming, if we're restricted to this open interval and that's okay because we're taking the limit as it approaches something within this open interval, and I'm being extra precise because I'm trying to explain it to you and it's important to be precise, but obviously, if you're working this out on a test or notebook, you wouldn't be taking, putting, or taking as much trouble to be putting all of these caveats in. So, what we've now realized is that, okay, this expression, actually, let's think about this. Let's think about the limit, the limit as theta approaches negative pi over 4 of this thing, without the restriction, of 1 over 1 minus the square root of 2 sine of theta. If we're dealing with this over, you know, with this open interval, wait, actually, even disregarding that, theta, this theta, or this expression is continuous at, it is defined and it is continuous at theta is equal to negative pi over 4 so this is just going to be equal to 1 over 1 minus the square root of 2 times sine of negative pi over 4. Sine of negative pi over 4. Sine of negative pi over 4, we've already seen is negative square root of 2 over 2, and so this is going to be equal to 1 over 1 minus square root of 2 times the negative square root of 2 over 2, so negative, negative, you get a positive, square root of 2 times square root of 2 is 2, over 2 is going to be 1. So this is going to be equal to 1/2. And so, I want to be very clear. This expression is not the same thing as this expression. They are the same thing at all values of theta, especially if we're dealing in this open interval except at theta equals negative pi over 4. This one is not defined and this one is defined, but as we've seen multiple times before, if we find a function that is equal to our original or an expression, is equal to our original expression, and all values of theta except, except where the original one was not defined at a certain point, but this new one is defined and is continuous there, well then these two limits are going to be equal, so if this limit is 1/2, then this limit is going to be 1/2, and I've said this in previous videos. It might be very tempting to say, well, I'm just going to algebraically simplify this in some way to get this, and I'm not going to worry about too much about these constraints, and then I'm just going to substitute negative pi over 4, and you will get this answer which is the correct answer but it's really important to recognize that this expression and this expression are not the same thing and what allows you to do this is, is the truth that if you have two functions, if you have f and g, two functions equal, let me write it this way, equal, equal for all x, except for all, wait, let me just write this this way, for all x except for a, then the limit, then, and let me write it this way, equal for all except, for all x except a and f continuous, continuous at a, then, then the limit of f of x as x approaches a is going to be equal to the limit of g of x as x approaches a, and I said this in multiple videos and that's what we are doing right here, but just so you can make sure you got it right, the answer here is 1/2.