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Tangent identities: periodicity

Sal solves a problem by considering the periodicity of the tangent function. Created by Sal Khan.

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  • leaf grey style avatar for user John Cashin
    So this whole time, we've been describing the slopes of the rays of these points on the unit circle with tan(x)? So the slope of the ray through 60 degrees is rad3/2/.5?
    (13 votes)
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  • leaf green style avatar for user 003honeydew
    is the terminal ray is the tangent of the subtended angle?
    (2 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      No, Sal's not talking about "tangent" as being a line (or ray), but about the trigonometric function (tan), which can be viewed as the slope of the terminal ray.

      There is, however, a reason to why the tangent function is called tangent.
      A ray, 𝑅, that emanates from the origin and forms the angle 𝜃 (0 < 𝜃 < 𝜋/2) with the 𝑥-axis intersects the unit circle in point 𝑇.
      The straight line that touches the unit circle in 𝑇 is a tangent of the unit circle.
      This line intersects the 𝑥-axis in a point 𝑋.
      It is also perpendicular to 𝑅, which means that the length of the line segment between 𝑇 and 𝑋 will be tan 𝜃.
      (6 votes)
  • starky ultimate style avatar for user Pranav K
    How do you do this without drawing a circle or without eyeballing it?
    (3 votes)
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    • orange juice squid orange style avatar for user Eugene
      By definition, an angle with the same slope is going to have the same tangent value. So, you don't have to visualize it on a unit circle in order to determine if two angles have the same tangent value.
      (1 vote)
  • blobby green style avatar for user Tobey
    Since tan ø = opp/adj = y/x and this is a unit circle I get how the numerator in tan 0.46 = 1/2 can be 1 but how can the denominator be 2?
    (0 votes)
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    • aqualine tree style avatar for user Judith Gibson
      Think of it this way.
      The tangent is a ratio of 2 sides of the triangle ( opposite / adjacent ).
      The 1 / 2 represents that ratio ,so, instead of representing actual lengths of the sides, it just indicates that the adjacent side is twice the length of the opposite side.
      So, for instance, the adjacent side could be .6 long and the opposite side then would be .3.
      (6 votes)
  • leaf green style avatar for user johnh9361
    Why is the tan of pi + 0.46 also a positive slope? If you start at the origin, isn't it then going downward since the arrow is going from right to left in a downward direction?
    (2 votes)
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    • male robot donald style avatar for user Soumya Acharya
      Yes but as you cross the pi angle, you enter into the 3rd quadrant, where the tangent of an angle is always +ve, which you can understand by :
      Recollecting that sine and cosine of an angle will always be of the same sign (-ve) in the third quadrant. or the first quadrant (+ve), or
      Imagining it as such that the line whose angle with the positive x-axis is a reflex angle, in counterclockwise sense, will come back to the first and third quadrant, similar to the initial position, assuming you slowly turn the line about the origin in a counterclockwise fashion, hence making the tangent of the angle effectively same. Therefore
      tan(pi+0.46)=tan(0.46)
      (3 votes)
  • area 52 yellow style avatar for user christiangurl2002
    my videos got stuck and wont let me finish or count the points for it. what should I do?
    (2 votes)
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  • blobby green style avatar for user William Mitchell
    The tan of (pi/2+0.46), (pi-0.46), and (2pi-0.46) is -(1/2).
    (2 votes)
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  • leaf green style avatar for user Scooley138
    This video seems like it's preparing the student for some practice problems. Where are those practice problems?
    (2 votes)
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  • boggle blue style avatar for user AProLearner
    At Sal mentions that with any degree, you can add pi and get the same tan 'function'. Will that work with radians since radians are another way to measure degrees?
    (1 vote)
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    • leaf green style avatar for user kubleeka
      Sal says that if you have an angle and add π to it, you get another angle with the same tangent. He is already talking about radians, since the problem at hand is in radians.

      If you were measuring angles in degrees, then adding 180º to any angle would give you another angle with the same tangent.
      (2 votes)
  • blobby green style avatar for user judy8nn
    Why is the first and the second one wrong? How can I know if the tangent is right
    (1 vote)
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Video transcript

Voiecover:One angle whose tangent is half is 0.46 radians. So we're saying that the tangent right over here is... So the tangent... So we're gonna write this down. So we're saying that the tangent of 0.46 radians is equal to half. And another way of thinking about the tangent of an angle is that's the slope of that angle's terminal ray. So it's the slope of this ray right over here. Yeah that makes sense that that slope is about half. Now what other angles have a tangent of 1 half? So let's look at these choices. So this is our original angle, 0.46 radians, plus pi over 2. If you think in degrees, pi is 180. pi over 2 is 90 degrees. So this one... Actually let me do in a color you're more likely to see. This one is gonna look like this. Where this is an angle of pi over 2. And just eyeballing it, you immediately see that the slope of this ray is very different than the slope of this ray right over here. In fact they look like they are. They are perpendicular because they have an angle of pi over 2 between them. But they're definitely not going to have the same tangent. They don't have the same slope. Let's think about pi minus 0.46. So that's essentially pi is going along the positive x axis. You go all the way around. Or half way around to your pi radians. But then we're gonna subtract 0.46. So it's gonna look something like this. It's gonna look something like that where this is 0.46 that we have subtracted. Another way to think about it, if we take our original terminal ray and we flip it over the y axis, we get to this terminal ray right over here. And you could immediately see that the slope of the terminal ray is not the same as the slope of this one, of our first one, of our original, in fact they look like the negatives of each other. So we can rule that one out as well. 0.46 plus pi or pi plus 0.46. So that's going to take us... If you add pi to this you're essentially going half way around the unit circle and you're getting to a point that is... Or you're forming a ray that is collinear with the original ray. So that's that angle right over here. So pi plus 0.46 is this entire angle right over there. And when you just look at this ray, you see its collinear is going to have the exact same slope as the terminal ray for the 0.46 radion. So just that tells you that the tangent is going to be the same. So I could check that there. And in previous videos when we explore the symmetries of the tangent function, we in fact saw that. That if you took an angle and you add pi to it, you're going to have the same tangent. And if you wanna dig a little bit deeper, I encourage you to look at that video on the symmetries of unit circle symmetries for the tangent function. So let's look at these other choices. 2 pi minus 0.46. So 2 pi... If this is 0 degrees, 2 pi gets you back to the positive x axis and then you're going to subtract 0.46. So that's going to be this angle right over here. And that looks like it has the negative slope of this original ray right up here. So these aren't going to have the same tangent. Now this one, you're taking 0.46 and you're adding 2 pi. So you're taking 0.46 and then you're adding 2 pi which essentially is just going around the unit circle once and you get to the exact same point. So you add 2 pi to any angle measure, you're going to not only have the same tangent value, you're gonna have the same sine value, cosine value because you're essentially going back around to the exact same angle when you add 2 pi. So this is definitely also going to be true.