Cosine is the derivative of sine.
y = sin(x) - Domain: D = ℝ, x, x0, x<> x0 in ℝ, Δx = x − x0, Δy = f(x)-f(x0).
[sin(x)] ’ = f’(x) = lim (Δy/Δx) = lim [f(x0+Δx)-f(x0)]/Δx = cos(x)          Δx→0          Δx→0   
The derivative of sine, y = sin(x) —by its conceptual definition as “slope of the tangent line”— is change-in-y-over-change-in-x = dy/dx = -sin(x)/1 = -sin(x)
Likewise, the derivative of sine is = cos.
Image - Shared at www.reddit.com/the_derivative_of_sine_is_cosine/
Zoom Info
Cosine is the derivative of sine.
y = sin(x) - Domain: D = ℝ, x, x0, x<> x0 in ℝ, Δx = x − x0, Δy = f(x)-f(x0).
[sin(x)] ’ = f’(x) = lim (Δy/Δx) = lim [f(x0+Δx)-f(x0)]/Δx = cos(x)          Δx→0          Δx→0   
The derivative of sine, y = sin(x) —by its conceptual definition as “slope of the tangent line”— is change-in-y-over-change-in-x = dy/dx = -sin(x)/1 = -sin(x)
Likewise, the derivative of sine is = cos.
Image - Shared at www.reddit.com/the_derivative_of_sine_is_cosine/
Zoom Info

Cosine is the derivative of sine.

y = sin(x) - Domain: D = ℝ, x, x0, x<> x0 in ℝ, Δx = x − x0, Δy = f(x)-f(x0).

[sin(x)] ’ = f’(x) = lim (Δy/Δx) = lim [f(x0+Δx)-f(x0)]/Δx = cos(x)
          Δx→0          Δx→0   

The derivative of sine, y = sin(x) —by its conceptual definition as “slope of the tangent line”— is change-in-y-over-change-in-x = dy/dx = -sin(x)/1 = -sin(x)

Likewise, the derivative of sine is = cos.

Image - Shared at www.reddit.com/the_derivative_of_sine_is_cosine/

Image 1: Multiplication of the integers modulo 15 - A beautiful illustration showing the emergence of primes and symmetry of multiplication. The colors were chosen to start blue at 1 (cold) and fade to red at n (hot). White is used for zero (frozen), because it communicates the most information about prime factorization.Image 2: Multiplication of the integers modulo 512.
The interactive version can be found here at ℤn Multiplication Visualizer βeta by Jared Deckard.Shared at: Math.stackexchange.com743542/1
Zoom Info
Image 1: Multiplication of the integers modulo 15 - A beautiful illustration showing the emergence of primes and symmetry of multiplication. The colors were chosen to start blue at 1 (cold) and fade to red at n (hot). White is used for zero (frozen), because it communicates the most information about prime factorization.Image 2: Multiplication of the integers modulo 512.
The interactive version can be found here at ℤn Multiplication Visualizer βeta by Jared Deckard.Shared at: Math.stackexchange.com743542/1
Zoom Info

Image 1: Multiplication of the integers modulo 15 - A beautiful illustration showing the emergence of primes and symmetry of multiplication. The colors were chosen to start blue at 1 (cold) and fade to red at n (hot). White is used for zero (frozen), because it communicates the most information about prime factorization.
Image 2: Multiplication of the integers modulo 512.

The interactive version can be found here at ℤn Multiplication Visualizer βeta by Jared Deckard.
Shared at: Math.stackexchange.com743542/1

Anonymous asked:

I admire your way donut cut into two halves. Have you noticed that pedestrians cross the road is not a straight line but exponentially, asymptotically tending to the opposite side, but not crossing it as long as possible. Maybe you'll have to talk about this as interesting.

Thanks for your opinion about the post!! It helps me think more about mathematical property of Möbius Strip in mathematics and in mathematical logic.

Thank you! :)

In the post, I did not create these images, I just collect and post them. 

image

Golden Ratio φ = (1+sqrt(5))/2 = 1.6180339887498948482&#8230;
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a &gt; b &gt; 0. Two quantities a and b are said to be in the golden ratio φ if
(a+b)/a = a/b = φ
One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ:
(a+b)/a = 1+ b/a = 1+1/φ
Therefore: 1+1/φ = φ  Multiplying by φ gives: φ^2 - φ - 1 = 0
Using the quadratic formula, two solutions are obtained:: 
φ = (1- sqrt(5))/2 or φ = (1+sqrt(5))/2
Because φ is the ratio between positive quantities φ is necessarily positive:
φ = (1+sqrt(5))/2 = 1.6180339887498948482&#8230;
See more at Golden Ratio.
Image: Phi (golden number) by Steve Lewis.

Golden Ratio φ = (1+sqrt(5))/2 = 1.6180339887498948482…

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0. Two quantities a and b are said to be in the golden ratio φ if

(a+b)/a = a/b = φ

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ:

(a+b)/a = 1+ b/a = 1+1/φ

Therefore: 1+1/φ = φ 
Multiplying by φ gives: φ^2 - φ - 1 = 0

Using the quadratic formula, two solutions are obtained::

φ = (1- sqrt(5))/2 or φ = (1+sqrt(5))/2

Because φ is the ratio between positive quantities φ is necessarily positive:

φ = (1+sqrt(5))/2 = 1.6180339887498948482…

See more at Golden Ratio.

Image: Phi (golden number) by Steve Lewis.