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Triangle similarity & the trigonometric ratios

Sal explains how the trigonometric ratios are derived from triangle similarity considerations. Created by Sal Khan.

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  • duskpin ultimate style avatar for user Ali Early
    What do you learn about a triangle from finding the sine, cosine and tangent?
    (76 votes)
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    • duskpin ultimate style avatar for user Artemis
      I wasnt sure that the other answers were really answering yours, they seemed to be more deep. If your question wasn't meant to be deep then I can answer it. When you take the sine, cosine, or tangent of a number you usually get a decimal number. Tangent is different, its usually a bigger number than the others. Now, this decimal number seems useless, i mean what do you do with it? Well, you can use this number to find a missing side length of a right triangle. Say you have all the angle measures but only one side length of a right triangle. You have the length of side A and you need to know the length of side B. Find which one you need sin, cos, or tan and enter it in the calculator. You get the answer so know you multiply the answer times the length of side A and the answer you get is side B!
      (25 votes)
  • leafers seedling style avatar for user Jasmine
    Can a cosine be negative? If so, when is it negative and when (if it can be positive) is it positive? Whenever I try to find cosines on my calculator, it is negative. Is that right?
    (39 votes)
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    • piceratops ultimate style avatar for user abassan
      A cosine can be negative if the angle is more than 90 degrees and less than 270 degrees.
      If you are using a calculator, you have to make sure it is set to degrees and not radians. If it is set to radians, you will get the wrong value all the time and you will sometimes get negatives when your answer should be positive.
      A simple check to see if your calculator is right is to take cos of 60 degrees. The answer should be 0.5 , if the calculator thought it was 60 radians the answer will be -0.95241298
      Hope this helps.
      (59 votes)
  • leafers seed style avatar for user Jessica Rajkumar
    Do the trigonometric definitions sine, cosine, and tangent apply to any angles of the right triangle? Meaning, can data be the 90 degree angle or can it only be one of the base angles?
    (4 votes)
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  • old spice man green style avatar for user Jonathan Ziesmer
    How do you determine the degree of an angle? Other than using a protractor! :D
    (8 votes)
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  • leaf blue style avatar for user Lyze Of Kiel 🟢Read bio I'm alive
    I feel like this is a dumb question, but what is theta?
    (4 votes)
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  • mr pink red style avatar for user scilock
    what is difference between similar and congruent triangles?
    (5 votes)
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  • duskpin ultimate style avatar for user C4LOwenZ
    Sal mentioned in the video that mathematicians gave trignometric ratios names: sine, cosine, and tangent. But of all the names they could've picked in the world, why those three? I don't see any connection between the ratios and the names.
    (3 votes)
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    • leaf green style avatar for user Tanner P
      Oddly enough, trigonometry is really about circles. And, as a result, the names for trig functions come from circles too.

      Sine comes from a Sanskrit word meaning "chord". A chord is a segment joining two points on the circumference of a circle.

      For cosine, the "co-" stands for complementary. Complementary angles are those that add to 90 degrees. If you take the cosine of an angle and the sine of its complement, you get the same answer. For example, cos(30)=sin(60).

      When talking about circles, a tangent is a line that hits one point on a circle. We still use the word "tangent" (besides the trig function tan) today, especially in calculus.

      There are three other trig functions that we don't use as often: cotangent, secant, and cosecant. A secant line is a line that intersects two points on a circle. Also, notice the pattern with the "co-". The complementary rule applies to tan/cot and sec/csc too.
      (9 votes)
  • leaf orange style avatar for user Golden pair & co
    could someone explain what sal is talking about please I don’t quite understand . is there another video i should watch to understand this because i’m new to trigonometry so i haven’t got a clue what i’m doing 😆
    (6 votes)
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    • hopper happy style avatar for user Hackcraft_
      In this video, Sal is explaining how to determine the relationship between two right triangles when they share an angle of the same measure (theta). They explain that if two triangles have two angles in common, the third angle is also the same. Since the sum of the angles of any triangle is 180 degrees, this means that the two triangles are similar. He then goes on to explain that the ratio of corresponding sides of similar triangles is always the same. Using this fact, he derive several equations relating the sides of the two triangles. These equations are true for any right triangle with an angle theta and are the trigonometric functions.
      (2 votes)
  • sneak peak blue style avatar for user HazixtheDruid
    Why are there not functions to calculate the ratios of angles other than 90°?

    e.g.
    sinₓ°(θ°) = opposite/hypotenuse of θ° in a x° triangle.
    cosₓ°(θ°) = adjacent/hypotenuse of θ° in a x° triangle.
    tanₓ°(θ°) = opposite/adjacent of θ° in a x° triangle.

    Here we could define hypotenuse as the angle opposite to x°, opposite as the side opposite to θ° and adjacent as the side adjacent to θ° that is not the hypotenuse.

    And this should work because of triangle similarity(Euclid's Elements, Book VI, Proposition 4):
    angle 1 = x°
    angle 2 = θ°
    angle 3 = 180-x°-θ°

    Establishing a relationship like this would help us solve for angles and sides in non-90° triangles. e.g.:
    x° = 60°
    θ° = 70°

    side adjacent to 70° = x
    side opposite to 70° = 5
    tan₆₀°(70°) = 5/x
    x = 5/tan₆₀°(70°)

    Thank you
    (4 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      Law of Sines takes care of that.
      sin(𝐴)∕𝑎 = sin(𝐵)∕𝑏 ⇒ 𝑎∕𝑏 = sin(𝐴)∕sin(𝐵)

      In other words, the ratio between any two sides in any triangle is equal to the ratio between the sines of their opposite angles.

      Given two angles, we easily calculate the third, and thereby we can find any trig ratio we want just using the sine function.

      In your example, the angle opposite to side 𝑥 is 180° − (60° + 70°) = 50°, and so
      5∕𝑥 = sin(70°)∕sin(50°) ⇒ 𝑥 = 5 sin(50°)∕sin(70°)
      (4 votes)
  • leaf green style avatar for user Harjo Winoto
    How is Sine 39 degree = 0.6293...? How is the scientific calculator makes this calculation?
    (4 votes)
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Video transcript

We've got two right triangles here. And let's say we also know that they both have an angle whose measure is equal to theta. So angle A is congruent to angle D. What do we now know about these two triangles? Well for any triangle, if you know two of the angles, you're going to know the third angle, because the sum of the angles of a triangle add up to 180 degrees. So if you have two angles in common, that means you're going to have three angles in common. And if you have three angles in common, you are dealing with similar triangles. Let me make that a little bit clearer. So if this angle is theta, this is 90. They all have to add up to 180 degrees. That means that this angle plus this angle up here have to add up to 90. We've already used up 90 right over here, so angle A and angle B need to be complements. So this angle right over here needs to be 90 minus theta. Well we could use the same logic over here. We already use of 90 degrees over here. So we have a remaining 90 degrees between theta and that angle. So this angle is going to be 90 degrees minus theta. You have three corresponding angles being congruent. You are dealing with similar triangles. Now why is that interesting? Well we know from geometry that the ratio of corresponding sides of similar triangles are always going to be the same. So let's explore the corresponding sides here. Well, the side that jumps out-- when you're dealing with the right triangles-- the most is always the hypotenuse. So this right over here is the hypotenuse. This hypotenuse is going to correspond to this hypotenuse right over here. And then we could write that down. This is the hypotenuse of this triangle. This is the hypotenuse of that triangle. Now this side right over here, side BC, what side does that correspond to? Well if you look at this triangle, you can view it as the side that is opposite this angle theta. So it's opposite. If you go across the triangle, you get there. So let's go opposite angle D. If you go opposite angle A, you get to BC. Opposite angle D, you get to EF. So it corresponds to this side right over here. And then finally, side AC is the one remaining one. We could view it as, well, there's two sides that make up this angle A. One of them is the hypotenuse. We could call this, maybe, the adjacent side to it. And so D corresponds to A, and so this would be the side that corresponds. Now the whole reason I did that is to leverage that, corresponding sides, the ratio between corresponding sides of similar triangles, is always going to be the same. So for example, the ratio between BC and the hypotenuse, BA-- so let me write that down-- BC/BA is going to be equal to EF/ED, the length of segment EF over the length of segment ED. Or we could also write that the length of segment AC over the hypotenuse, over this triangle's hypotenuse, over AB, is equal to DF/DE-- once again, this green side over the orange side. These are similar triangles. They're corresponding to each other. So this is equal to DF/DE. And we could keep going, but I'll just do another one. Or we could say that the ratio of this side right over here-- this blue side to the green side of this triangle-- the length of BC/CA is going to be the same as the ratio between these two corresponding sides, the blue over the green, EF/DF. And we got all of this from the fact that these are similar triangles. So this is true for any right triangle that has an angle theta. Then those two triangles are going to be similar, and all of these ratios are going to be the same. Well, maybe we can give names to these ratios relative to the angle theta. So from angle theta's point of view-- I'll write theta right over here, or we can just remember that-- what is the ratio of these two sides? Well from theta's point of view, that blue side is the opposite side. It's opposite-- so the opposite side of the right triangle. And then the orange side we've already labeled the hypotenuse. So from theta's point of view, this is the opposite side over the hypotenuse. And I keep stating from theta's point of view because that wouldn't be the case for this other angle, for angle B. From angle B's point of view, this is the adjacent side over the hypotenuse. And we'll think about that relationship later on. But let's just all think of it from theta's point of view right over here. So from theta's point of view, what is this? Well theta's right over here. Clearly AB and DE are still the hypotenuses-- hypoteni. I don't know how to say that in plural again. And what is AC, and what are DF? Well, these are adjacent to it. They're one of the two sides that make up this angle that is not the hypotenuse. So this we can view as the ratio, in either of these triangles, between the adjacent side-- so this is relative. Once again, this is opposite angle B, but we're only thinking about angle A right here, or the angle that measures theta, or angle D right over here-- relative to angle A, AC is adjacent. Relative to angle D, DF is adjacent. So this ratio right over here is the adjacent over the hypotenuse. And it's going to be the same for any right triangle that has an angle theta in it. And then finally, this over here, this is going to be the opposite side. Once again, this was the opposite side over here. This ratio for either right triangle is going to be the opposite side over the adjacent side. And I really want to stress the importance-- and we're going to do many, many more examples of this to make this very concrete-- but for any right triangle that has an angle theta, the ratio between its opposite side and its hypotenuse is going to be the same. That comes out of similar triangles. We've just explored that. The ratio between the adjacent side to that angle that is theta and the hypotenuse is going to be the same, for any of these triangles, as long as it has that angle theta in it. And the ratio, relative to the angle theta, between the opposite side and the adjacent side, between the blue side and the green side, is always going to be the same. These are similar triangles. So given that, mathematicians decided to give these things names. Relative to the angle theta, this ratio is always going to be the same, so the opposite over hypotenuse, they call this the sine of the angle theta. Let me do this in a new color-- by definition-- and we're going to extend this definition in the future-- this is sine of theta. This right over here, by definition, is the cosine of theta. And this right over here, by definition, is the tangent of theta. And a mnemonic that will help you remember this-- and these really are just definitions. People realized, wow, by similar triangles, for any angle theta, this ratio is always going to be the same. Because of similar triangles, for any angle theta, this ratio is always going to be the same. This ratio is always going to the same. So let's make these definitions. And to help us remember it, there's the mnemonic soh-cah-toa. So I'll write it like this. soh is sine is opposite over hypotenuse. cah-- cosine is adjacent over hypotenuse. And then finally, tangent is opposite over adjacent-- soh-cah-toa. And in future videos, we'll actually apply these definitions for these trigonometric functions.