# 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. c4talin94 says:

That was interesting. Thank you!

2. Duncan Carr says:

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!

3. LTS1287 says:

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.

fantastic point of view .

5. Deltoidz says:

This is a great explanation

6. Thom Gibson says:

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!

7. wellbornwinter6 says:

why it was invented & what was its application in the very beginning?

8. Ally NKUNDIBIZA says:

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

9. Andrew Peal says:

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

10. N Dilip Kumar says:

awesome sir awesome thank you thank you very much

11. Rockistar says:

Great explanation – but why would you use inches!

12. 뀨돌이 says:

Greek thy had known bout negative numbers?????before 3000 years ago?

13. Jesse McElroy says:

That was a terrific video!

14. Daniel Ram says:

seriously, thank you

15. TABREZ SHAMS HASHMI says:

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

16. sebastian cuello says:

I'm finding this out when I already finished trigonometry and calculus

17. XKL 1010 says:

Lo and behold!
Thanks for the vid

18. narendra kumar reddy padala says:

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

19. vijaysingh chauhan says:

It was just awesome??????????

Thank you sir

21. Biniam Tesfay says:

You did it Great

22. Biniam Tesfay says:

But why u use inches why not Cm?

23. bhupal singh satyapal says:

Thanku sir, I got understood very well. Keep making viedo and keep things easy. ?

24. Jereme Salazar says:

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.

25. Babak Aghayarov says:

great explanation. thanx a lot. what is the name of teacher?

26. Marulasiddappa H E says:

amazing!!!!!!!! so easy to understand !!!! your teaching is so interesting. thank you !!!!!!!!!

27. AB says:

Thank you sir, well explained

28. Rachel Harris says:

Omg why are you working in inches what is wrong with you?!

29. Cole Smith says:

I would argue that this is how to use the unit circle. I was trying to find more information on the series.

30. Lukesh30253 says:

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

31. Beeg Yoshi says:

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)

This is just unit circle/sohcahtoa though

33. sahir khan says:

Sir
Y this ratio is named sine

34. Mortified Heine says:

Super cool!

35. Arash Chitgar says:

great explanation. thanks a lot!

36. Himanshu Sahu says:

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

37. J dawg says:

Amazing explanation! Now i finally understand how the functions actually work

38. Chrono826 says:

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

Thank you! I was looking for understanding on this!

39. Robert Hughes says:

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!

40. Sani Hyne says:

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?

41. Bill says:

What a great way to explain the sine function!

42. Ikpe Essien says:

honesstly, it is today that i understand and know the origin of sine function.

43. Subidita Chakraborty says:

Very clearly explained. I got a great lesson.Thank you very much

44. Nirmal Babu says:

Did trigo 6 years ago. Now i know how the sin function actually works

45. John Days says:

I like how educated people explain stuff but only they can hardly understand their own selves..

46. Awaseme says:

What kind of scientist measures in inches?

47. Santhosh says:

but why draw a triangle within a circle? what's the use case for it?

48. Ashok Tiwari says:

Wonderful video sirr????

49. kannan *-* says:

Amazing discription.. ??

50. Mateusz Wysocki says:

Thank you Sir for your good work

51. MageBurger says:

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!

52. Alberto M. A. says:

but how does it work internally? you can't measure the side and get a high precision

53. Max Dodge says:

Black shirt and white collar make me feel like I'm being taken to the church of trigonometry.

54. VIRGIN UNIVERSE - says:

https://youtu.be/2H9ileO4u5Y

55. Co0lguy27 says:

this is cool and all but what does the sine represent and why does it exist

56. Krishna Teja Chakrapu says:

Awesome explaination… He deserve googol googol likes.

57. tommy Bull says:

FFFFFFF School…

These teachers are horrible.

This is so clear and simple.