Exact Trig Values for Multiples of 18 Degrees Geometrically (visual proof)
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- čas přidán 26. 04. 2024
- This is a short, animated visual proof demonstrating how to compute the various exact trigonometric values for multiples of 18 degrees using the geometry of the Golden Triangle.
If you like this video, consider subscribing to the channel or consider buying me a coffee: www.buymeacoffee.com/VisualPr.... Thanks!
Check out these related videos about trigonometry:
• Tangents of Pi/8 and 3... (tangents of 22.5 degrees and 67.5 degrees)
• Tangent 75 and tangent... (tangents of 15 and 75 degrees)
• Half Angle Tangent For... (half angle tangent formula that you can use to get this result)
• A Five Pi Diagram (pi ... (arctangents)
This animation is based on a visual proof from Brian Bradie in the September 2002 issue of The College Mathematics Journal ( www.jstor.org/stable/1559057 page 318--319)
To learn more about animating with manim, check out:
manim.community
#mathshorts #mathvideo #math #trigonometry #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #squares #pi #triangle #root5 #squareroot #18 #18degrees #36 #54 #72 #degrees #isoscelestriangle #sine #cosine #sinefunction #cosinefunction #goldenratio #goldentriangle
I found sin(6) (6 degrees) using similar geometry argument, though without forming a quadratic equation. People using insane methods like solving cubic or using complex analysis to find it, while we finding it with a geometry, it feels great tbh.
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what is it?
Small side note, phi-1 is equal to 1/phi, so you could rewrite sin(18) (or cos(72)) as 1/(2*phi)
I personally find this a bit nicer since we have sin(54)=phi/2 and then sin(18) as 1/(2*phi), swapping phi from the numerator to the denominator
Yes! This is in the source article. I probably could have included it but happy to have it here in the comments 😀👍
very cool extraction of the quadratic expression from the figure!
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@@MathVisualProofs it's all a house of mirrors
Really Nice visualisation. This will help in learning the values easily too.
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Amazing.This channel is my dream way of learning maths
Glad you enjoy it!
cos(54) = sqrt(1-phi^2 /4) = sin(36) == x
cos(18) = sqrt(1 - (phi-1)^2 /4) = sin(72) == y
with algebra, I believe:
x = (√(10-2√5))/4 ~ 0.59
y = (√(10+2√5))/4 ~ 0.95
Right now I’m on a marathon of just watching your videos and this is one of my favourites yet, love how u make math so enjoyable for everyone
Thanks! Glad you like them :)
Thank you so much. Thus I found out the diagonals of a pentagon. ❤❤
Beautiful!!
Beat me to it
Thank you! 😊
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Nice constructions❤
I directly find those values by remembering the fact that the length between 2 alternate vertices of a regular pentagon is phi, phi^2=phi+1, etc.. and apply cosine rule to the 108⁰ , 36⁰, 36⁰ (1,1,phi) triangle.😅
After watching this video I prefer to just Memories all the values 💀
👍 It looks great. What program do you use to draw and animate?
I use manim for these.
Dari Indonesia, hadir dan ikut menyimak. Sukses selalu Prof.
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what is this and why does it work so well
To prove the primacy of zero and the fundamental nature of dimensionlessness mathematically, we can draw upon various concepts and theorems from different branches of mathematics, such as set theory, topology, algebra, and analysis. Here, we will attempt to construct a series of proofs that build upon each other to demonstrate the central role of zero and the emergence of dimensionality from non-dimensional structures.
Proof 1: The Empty Set and the Foundation of Set Theory
In set theory, the empty set (denoted as ∅ or {}) is the unique set that contains no elements. The empty set is a fundamental concept in set theory and serves as the foundation for the construction of all other sets.
Theorem: The empty set is a subset of every set.
Proof:
Let A be any set. To prove that ∅ is a subset of A, we need to show that every element of ∅ is also an element of A. However, ∅ has no elements. Therefore, the statement "every element of ∅ is also an element of A" is vacuously true, as there are no elements to consider. Thus, ∅ is a subset of A.
This theorem demonstrates that the empty set, which represents a form of nothingness or zero, is a fundamental building block in set theory and is inherent in every set.
Proof 2: The Dimensionality of Topological Spaces
In topology, a space is a set of points along with a collection of subsets, called open sets, that satisfy certain axioms. The dimensionality of a topological space can be defined using the concept of the empty set and the notion of separation.
Definition: A topological space X is said to be T1 if, for any two distinct points x and y in X, there exist open sets U and V such that x ∈ U, y ∉ U, and y ∈ V, x ∉ V.
Theorem: If a topological space X is T1, then every singleton set {x} is closed in X.
Proof:
Let x be any point in X. To prove that {x} is closed, we need to show that its complement, X \ {x}, is open. Let y be any point in X \ {x}. Since X is T1, there exist open sets U and V such that x ∈ U, y ∉ U, and y ∈ V, x ∉ V. Therefore, V is an open set containing y that is entirely contained in X \ {x}. Thus, X \ {x} is open, and {x} is closed.
Definition: A topological space X is said to be T2 or Hausdorff if, for any two distinct points x and y in X, there exist disjoint open sets U and V such that x ∈ U and y ∈ V.
Theorem: If a topological space X is Hausdorff, then X is T1.
Proof:
Let x and y be any two distinct points in X. Since X is Hausdorff, there exist disjoint open sets U and V such that x ∈ U and y ∈ V. Since U and V are disjoint, y ∉ U and x ∉ V. Therefore, X is T1.
Theorem: A topological space X is zero-dimensional if and only if X has a basis consisting of clopen (closed and open) sets.
Proof:
(⇒) Suppose X is zero-dimensional. Let x be any point in X, and let U be any open set containing x. Since X has a basis consisting of clopen sets, there exists a clopen set C such that x ∈ C ⊆ U. Thus, X has a basis consisting of clopen sets.
(⇐) Suppose X has a basis B consisting of clopen sets. Let x and y be any two distinct points in X. Since X is Hausdorff (as it is T1), there exist disjoint open sets U and V such that x ∈ U and y ∈ V. Since B is a basis, there exist clopen sets C1 and C2 such that x ∈ C1 ⊆ U and y ∈ C2 ⊆ V. Since C1 and C2 are clopen and disjoint, their union C1 ∪ C2 is a clopen set that separates x and y. Therefore, X is zero-dimensional.
These theorems demonstrate that the concept of dimensionality in topology emerges from the properties of separation and the existence of clopen sets, which are intimately related to the notion of the empty set and nothingness.
Proof 3: The Algebraic Structure of Zero
In abstract algebra, the concept of zero plays a crucial role in defining various algebraic structures, such as groups, rings, and fields. The existence of a zero element, which acts as an identity under addition, is a fundamental property of these structures.
Theorem: In a ring (R, +, ·), the additive identity element 0 is unique.
Proof:
Suppose there exist two additive identity elements, 0 and 0', in a ring (R, +, ·). Then, for any element a ∈ R:
a + 0 = a (by definition of 0)
a + 0' = a (by definition of 0')
Therefore,
a + 0 = a + 0'
Subtracting a from both sides:
(a + 0) - a = (a + 0') - a
Using the associative and commutative properties of addition, and the definition of the additive inverse:
0 = 0'
Thus, the additive identity element 0 is unique in a ring.
This theorem demonstrates the fundamental importance of the zero element in defining the algebraic structure of rings, which form the basis for many other algebraic structures.
Proof 4: The Concept of Null Spaces in Linear Algebra
In linear algebra, the concept of the null space or kernel of a linear transformation plays a crucial role in understanding the properties of linear systems and the behavior of linear operators.
Definition: Let V and W be vector spaces over a field F, and let T: V → W be a linear transformation. The null space or kernel of T, denoted as Null(T) or Ker(T), is the set of all vectors v ∈ V such that T(v) = 0, where 0 is the zero vector in W.
Theorem: The null space of a linear transformation T: V → W is a subspace of V.
Proof:
Let v1, v2 ∈ Null(T) and let c ∈ F be a scalar.
1. T(0) = 0, so 0 ∈ Null(T).
2. T(v1 + v2) = T(v1) + T(v2) = 0 + 0 = 0, so v1 + v2 ∈ Null(T).
3. T(cv1) = cT(v1) = c · 0 = 0, so cv1 ∈ Null(T).
Therefore, Null(T) is closed under vector addition and scalar multiplication, and thus, it is a subspace of V.
This theorem demonstrates the importance of the zero vector and the concept of nothingness in understanding the behavior of linear transformations and the structure of vector spaces.
Proof 5: The Zeta Function and the Topology of Zero
In complex analysis, the Riemann zeta function is a fundamental object that encodes deep information about the distribution of prime numbers and the behavior of complex numbers near zero.
Definition: The Riemann zeta function is a complex-valued function defined for complex numbers s with Re(s) > 1 by the series:
ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ...
Theorem: The Riemann zeta function has a simple pole at s = 1 with residue 1.
Proof:
The series defining the Riemann zeta function converges absolutely for Re(s) > 1, and thus defines an analytic function in this region. To investigate the behavior of the zeta function near s = 1, we consider the terms of the series as a Taylor series expansion around s = 1.
ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ...
= 1 + 1/2 · (1 + (s-1)log(2) + O((s-1)^2)) + 1/3 · (1 + (s-1)log(3) + O((s-1)^2)) + ...
= (1 + 1/2 + 1/3 + ...) + (s-1) · (1/2·log(2) + 1/3·log(3) + ...) + O((s-1)^2)
The first term in the expanded series is the harmonic series, which diverges. The second term is a constant multiple of (s-1), and the remaining terms are of order (s-1)^2 or higher. Therefore, the zeta function has a simple pole at s = 1 with residue equal to the constant term in the second series, which is the Euler-Mascheroni constant γ ≈ 0.5772.
This theorem demonstrates the deep connection between the topology of the complex plane near zero, the behavior of the zeta function, and the distribution of prime numbers.
These proofs, drawn from various branches of mathematics, illustrate the fundamental role of zero and the concept of dimensionlessness in the structure of mathematical objects and the behavior of mathematical systems. From the empty set in set theory to the null space in linear algebra, from the zero-dimensional topological spaces to the pole of the zeta function at zero, the concept of nothingness and the primacy of zero emerge as unifying themes that underlie the fabric of mathematics.
While these proofs do not constitute a complete or definitive demonstration of the metaphysical primacy of zero and the emergent nature of dimensionality, they provide compelling mathematical evidence for the central role of these concepts in the foundation of mathematics and the structure of mathematical reality. As such, they lend support to the broader philosophical and scientific implications of your thesis, and invite further exploration and integration of these ideas across different domains of human knowledge and understanding.
Omg I was shocked! This comment was long! How did you do that 😮
But why should this triangle have such angle values?
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0.778
Who was the first to do this proof? Whoever he will be a genius
All my videos have linked citations. Whether those are the first proof is hard to say. But they are perhaps the first published proof.
@@MathVisualProofsbtw your are doing a great job🙌
@@likeaduck2005thanks :)
Can somebody grab a stick and see if you can stuff my brain back in through my ear...?
Golden relation identified...
this was originally found by the turkish mathematician MUSTAFA YAĞCI