Origin of the Sine Function Part 1

Origin of the Sine Function Part 1


Where does the sign function come from we’ve all seen this function graphed before? But where does it come from so I’ve plotted this graph in terms of degrees for demonstration purposes? To understand whether where the sine function comes from we need to take a look at a circle? I’ve plotted a circle here with a radius of 12 inches and Inside of the circle I’ve inscribed triangles each of the triangles has a ninety degree Angle at its foot And I’ve plotted them at different at different degrees So this first one has an angle of 30 degrees relative to the x-Axis this one here and then this one here has an angle of 45 degrees relative to the x-Axis and Then this one here has an angle of 60 degrees relative to the x-Axis so Where the sine function comes from the sine function comes from plotting the height of each of these triangles? Divided by the radius that is the hypotenuse of each triangle the hypotenuse of each triangle is the radius of the circle, right? Because this is 12. This is 12 this is 12, so the hypotenuse is always the same for each of them and the sine function comes from dividing the height of each triangle divided by that hypotenuse, so Let’s do the first one for 30 degrees. We will look at this triangle right here Let’s measure the height here, and we the height come is 6 inches So 6 divided by 12 is 0.5. So we come over here. We look at our graph, and we come out to 30 degrees and We see that 0.5. Is on the graph, Lo and Behold So we contend that the sine of theta is simply equal to the ratio of the height of each respective Triangle divided by the hypotenuse of each respective triangle, which is the radius so this is simply equal to Y over R so then let’s do it for the next one so this one this one has an angle of 45 degrees relative to the x-Axis and We measure the height of it and the height of it is eight point four eight inches So eight point four eight divided by 12 is equal to 0.7 zero seven so we come over here 45 degrees And we come up to the graph and Lo and Behold It’s at a it’s at an elevation of 0.7 zero seven it has a y-value of point seven or seven And then we come up, we would take a look at the if the triangle corresponding to sixty degrees We measure the height of it and we get a height of ten point three nine Ten point three nine inches ten point three nine inches divided by 12 is equal to 0.8 six six So we come out to sixty degrees we come up to the graph and Lo and behold point eight six six is right there And of course we see the number one because at ninety degrees It’s just the radius divided by the radius because in ninety degrees the height of this triangle is is the height of the radius because we see that the the width of this triangle is getting shorter and shorter and children until right here when the width of the triangle is equal to 0 And we progress around the circle plotting this ratio for all these different triangles The one catch is that when we get to 180 degrees We go negative Because all of these triangles were above the x-Axis so these all had positive y values positive heights that is but all of these have negative heights because they’re below the x-Axis, so We see that from 0 to 180 we have positive values And we just plot all the way around the circle But then when we get to 180 we go negative, and we have negative values That is the ratios or Nega so let’s just take a look at one of those Take a look at this triangle right here this triangle has an angle of 210 degrees So it’s 180 plus 30 which is 210 and so we measure the height of that one and The height is 6 but it’s below the x-Axis so it’s negative 6 negative 6 divided by 12 is negative 0.5 so, Lo and behold we come out to 210 degrees you come down to the graph and The whole look it’s right there negative 0.5. Negative 0.5 And we continue plotting around the circle to get the rest of these values That’s it that is where the sine function comes from in the next tutorial we are going to discuss What the utility of the sine function is and you know how can we use this thing? How can we exploit this thing to to our advantage okay? See you next time

57 Replies to “Origin of the Sine Function Part 1”

  1. I've never seen a tutorial on the sine function before. I bet if I asked 1,000 people to explain it that 999 would fail. Another awesome tutorial that everyone should watch. Thank you!

  2. I don't understand why I wasn't taught this in high school. They make the sine function look like wizardry when its actually really simple. Thank you for this video.

  3. Man these visuals are huge. I remember all of the rules and ratios with all the trig functions but never really knew what they were saying how it related to circles or anything. Excellent!

  4. I really appreciate your help. But I still don't understand how the calculator is given
    a value (for example 0.707), and it computes whatever trigonometrical function value the user wants. For example arcsin(0.707) = 44.99

  5. I wish videos like this were around when I was in school. I took trig in high school and remember asking where these functions come from and my teacher looked at me like I was speaking another language. I hated math than but as an adult I find it eloquent and beautiful. Thanks for the awesome video

  6. Sir,
    I am confused
    Because as we consider the sign of y according to y axis as positive or negative,
    When why not we consider the sign convention for radius ie which is representing x axis

  7. Thank you sir…its really a nice explaination..i got very useful points from this lecture

  8. Thanks for this explanation. This is a good background study for the student in order for them to visualize the use and function of the lines in the graph. I too got something from this clips…very nice, I like it.

  9. I’m a tile setter, if you can easily graph out cool stuff like that you should go make waterjet mosaics and make some $

  10. but why did you make the radius of the circle 12" when you could have made a unit circle so the hypotenuse is 1 so the opposite of the triangle is sin(θ) and the adjacent would be cos(θ)

  11. How sin¢=y/R comes because you are taking sinx as height divided by hypotenuse . Could u please tell me how this is formulated

  12. Sine of theta = position of curvature relative to its radius.

    Thank you! I was looking for understanding on this!

  13. This is brilliant, i've searched a few different videos for a good, thorough explanation and this is by far the best i've seen. Well done!

  14. Hi boss. Great explanation.I am new to mathematics & i have a one question. What if we increase the radius from 12 to 20? will sin(30) always return 0.5? if yes, please elaborate how? As per my understanding. if we change the radius then on angle 30, height will be different to get 0.5. is it?

  15. Mind. Flipping. Blown.

    And I just stumbled here out of pure curiosity (also because my friend in my Physics class expressed genuine frustration for not knowing what sine actually meant).

    Thanks!

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