Very easy to solve if you know that sin x < x for any positive real x and if you note that sin(π/10) = sin 18° = cos 72° = ¼(√5 − 1) so we have ½(√5 − 1) = 2·sin(π/10) < 2·π/10 = π/5
You probably don't need to review so much basic information in these videos. I think everyone knows what a unit circle and sin and cosine are. But a little more explanation of less obvious things, like how you know cos(pi/5) = (sqrt(5) + 1)/4 , would be helpful.
The ancient Egyptians and Mesopotamians, including the Babylonians, had knowledge of pi () long before it was formally defined. The Egyptians approximated to about 3.16, as seen in the Rhind Papyrus, which dates back to around 1650 BC. The Babylonians, during the Old Babylonian Period (circa 1900-1680 BC), used a value of as 3 and later refined it to 3.125. These approximations were crucial for their architectural and mathematical achievements, despite being less accurate than later calculations by the Greeks and Chinese
9:00 I do not get the least bit what you mean here. I could easily follow the previous sections, but exactly at this point you unnecessarily speed up a lot. Why the heck are cosine(36) and cosine(37) "conjugates", and what does "conjugate" mean in this context (I know complex conjugates of course)? And why is cos(pi/5) = (1 + sqrt(5))/4? Do you know this from a formula book?
2.2²=2(2.4)+0.2²=4.84 y 2.25²=2(2.5)+0.25²=5.0625 con 25²=625 Luego 2.25(-1+√5) de donde al dividir entre 10 en ambos miembros se obtiene que: π/5 > (-1+√5)/2.
Fórmula utilizada: a²-b²=(a-b)(a+b) en la forma equivalente a²=(a-b)(a+b)+b² con a=2.2 y b=0.2 para el primer número y con a=2.25 y b=0.25 para el segundo número.
![Hunxho - Ball Forever [Official Video]](http://i.ytimg.com/vi/nki9ZTF9Uls/mqdefault.jpg)
pi/5 vs. (sqrt(5) - 1)/2
10[pi/5] vs. 10[(sqrt(5) - 1)/2]
2*pi vs. 5sqrt(5) - 5
(> 2*3.14) vs. [< 5*sqrt(81/16) - 5]
(> 6.28) vs. [< 5*(9/4) - 5]
(> 6.28) vs. [< 5*2.25 - 5]
(> 6.28) vs. [< 11.25 - 5]
(> 6.28) *>* [< 6.25]
Therefore, pi/5 > (sqrt(5) - 1)/2
Pi>3.13 so pi/5>0.625 ,but (sqrt5-1)/2
👏👏👏👏👏👏
Nice!
I’m disappointed that nobody recognized the second one as phi-1 or 1/phi and just multiplied 0.618 by 5 to get 3.09. Thus, 1/phi
Cool - you sort of made this into a geometry puzzle, your favorite pastime!
Indeed!
Very easy to solve if you know that
sin x < x
for any positive real x and if you note that
sin(π/10) = sin 18° = cos 72° = ¼(√5 − 1)
so we have
½(√5 − 1) = 2·sin(π/10) < 2·π/10 = π/5
We know pi/10 >3.14/10
( √5 -1)/2 is multiplied by (√5+1)/2 We get product 1
(√5 +1)/2 >3.23/2
(3.14*3.23)/10=1.011~
pi/5>(√5-1)/2
You probably don't need to review so much basic information in these videos. I think everyone knows what a unit circle and sin and cosine are. But a little more explanation of less obvious things, like how you know cos(pi/5) = (sqrt(5) + 1)/4 , would be helpful.
Probably
The second number is the reciprocal of the golden ratio.
|BC| = versine(alpha) = 1 - cosine(alpha); |AC| = crd(alpha) = 2*sine(alpha/2)
Cool,cool😊😊😂❤
The ancient Egyptians and Mesopotamians, including the Babylonians, had knowledge of pi () long before it was formally defined. The Egyptians approximated to about 3.16, as seen in the Rhind Papyrus, which dates back to around 1650 BC. The Babylonians, during the Old Babylonian Period (circa 1900-1680 BC), used a value of as 3 and later refined it to 3.125. These approximations were crucial for their architectural and mathematical achievements, despite being less accurate than later calculations by the Greeks and Chinese
So clearly Greeks didn't discover pi. Hopefully this will accurate education on the history of mathematics in general and PI in particular.
3.1416/5=0.62832
(2.236-1)/2=1.236/2=0.618
9:00 I do not get the least bit what you mean here. I could easily follow the previous sections, but exactly at this point you unnecessarily speed up a lot. Why the heck are cosine(36) and cosine(37) "conjugates", and what does "conjugate" mean in this context (I know complex conjugates of course)? And why is cos(pi/5) = (1 + sqrt(5))/4? Do you know this from a formula book?
Fantastic...❤❤❤.
Thanks 🤗
👌👏👏👏
2.2²=2(2.4)+0.2²=4.84 y 2.25²=2(2.5)+0.25²=5.0625 con 25²=625
Luego 2.25(-1+√5) de donde al dividir entre 10 en ambos miembros se obtiene que:
π/5 > (-1+√5)/2.
Fórmula utilizada: a²-b²=(a-b)(a+b) en la forma equivalente a²=(a-b)(a+b)+b² con a=2.2 y b=0.2 para el primer número y con a=2.25 y b=0.25 para el segundo número.
I know that π/5= 3.14/ 5= .628
N √5= 2 236 so √5-1/ 2= .66
Now obvios is soln.
Your second line is wrong. You need grouping symbols around sqrt(5) - 1.
@robertveith6383 use. Ur mind
π=3.14 so π/5 =.628
but sqrt 5 is 2.23
So (2.23 -1)/2 gets .615
The former beats.
No, you are using approximations. Anyway, sqrt(5) is closer to 2.24.