This reminds me of the time my math teacher was teaching us some graphing function and the line he drew barely missed a point so he just made the point bigger so it connected
It has been hard enough for me to accept the fact that the limit of the ratio, of any 2 sequential terms in the Fibonacci series, equaled The Golden Ratio. Now, I find the Golden Ratio is also 2 x sin (pi/10). This is almost as cool as Euler's equation. Many thanks!
Again, a great video. In high school, we learned the method of taking 5θ = 90° and doing tedious manipulations. This method is so much better and pure.
Inscribe a regular pentagon in a circle or radius 1, the vertexes are (1,0), (cos 72, sin 72),(cos 144, sin 144), (cos 72, - sin 72), (cos 144, - sin 144). The average of these coordinates will be the center of the pentagon (0,0). 1 + 2 cos 72 + 2 cos 144 = 0 1 + 2 cos 72 + (2 cos^2 72 - 1)= 0 4 cos^2 72 + 2 cos 72-1 = 0 Apply the quadratic formula cos 72 = -1/4 + sqrt(5) / 4
Dear Sir, I really appreciated your efforts in this wonderful piece of geometry explanation. I really liked this new way of solving and obtaining the. Golden ratio... Sir,. I discovered that 18°*5 = 90 degrees therefore I found the value of sin18° by algebra and so I wanted to share it with you .. Let 18° =x 2x+3x =90° 2x= 90°- 3x Taking the sine on both sides, You obtain Sin(2x)= sin (90- 3x) 2sinx cosx = cos 3x Sir .. after solving these equations by eliminating cos(x) from both sides and converting cos^2x into sin^2x we notice that a quadratic equation is formed. We solve it and obtain the same value as you did using this amazing mind bending geometry , I.e( (5)^0.5 - 1)/4 .
You have to know, however, how to write cos(3x) differently. I checked that it's equal to cos^3(x) - 3sin^2(x)*cos(x), but that's not something I would know on top of my head. Once you know that, however, it is really nice way!
Abdullah Kanee Thanks :) Yes, in my maths class in high school we didn't learn triple angles identities, and since then, I was mainly using numerical values for the angles (or sinx=x approximation ;) ). In general, sin(2x), cos(2x), as well as sin(x+y), sinx+siny etc. are probably more useful. But it's easy to forget that there are other possibilities, if you don't use them :D
72 18 90 right triangle is useful in regular pentagon construction Hypotenuse of this triangle has length a lim_{n\to \infty} \frac{F_{n+1}}{F_{n}} where F_{n} is nth Fibonacci number
The moment i get convinced you are the true math ninja... U hit me with a blow so powerful that i realize how my previous notion was such an obscene understatement.
Thanks for the proof And i'll do a copy pasta math joke Q:Why don't you accept people to drink in your math party? A:Cuz you cant drink and derive (sorry :D)
This is one of the very few youtube maths videos where I can honestly say; been there, done that got the (Fibonacci ) Tee shirt. When I was 8 my teacher showed me a compass and straightedge construction of a regular pentagon in a given circle. 30 years later I was able to prove the construction using that triangle.
There was a much faster way of getting to that quadratic equation, why didn't you do it this way? the triangles are similar so corresponding sides are proportional 1/x=x/mystery number cross multiply x^2=mystery number 1=x+mystery number aka x^2 x^2+x-1=0
Here is another interesting fact. If you draw a regular Pentagon and join all the diagonals, you get a smaller Pentagon inside. The ratio of the side of the bigger Pentagon to the side of the smaller Pentagon is golden ratio^2. Also sine(666) = -(golden ratio)/2 #sacredgeometry
Simply a perpendicular bisector. That is a simple ratio given by Euclid in Data as well as elements. It is also taught in Geometry. As well as a modified G-conjecture.
This is a very quick result but the method of constructing a rectangle from two differently sized 45° right triangles, one 30-60-90° right triangle and the 18-72-90° (rt. ) triangle is elegant & beautiful
4:06, at this point you could say that the second triangle has a missing side of x^2 since the triangle was changed by a factor of x from the first one. Then at 6:03 you notice it's the same as 1-x. Then your equation becomes clear that x^2 = x-1 and get your solution
This is equal to (1/2) * phi^(-1). The cosecant is equal to 2 * the golden ratio. In the original 36-72-72 triangle the ratio of the sides is exactly the golden ratio. This is not so surprising because the golden ratio is (1 + sqrt(5))/2 and all the angles of the triangle are measured in fifths of 180 degrees (for the isosceles one).
Actually there is big Phi and small phi so it might be even closer than you think. Your explaination though doesn't make immediately sense to me but I'm sure if I thought a bit about it, it would!
Ok. I belive it is not trivial that (sqrt(5)-1)/2 = -(1/2) * (1-sqrt(5))/2 or true. Minor mistake perhaps have been added for this comment. But what is also not trivial and that I believe to be even more impressive as far as I understand, major phi(golden ratio) equals 1 over minor phi(other golden ratio) and also 1 plus minor phi(other golden ratio).
We easily construct angle if tangent can be expressed with four arithmetic operations and taking square roots Addition and subtraction can be realized by moving segments with compass , multiplication and division can be realised with Thales' theorem square root we will get after geometric mean with unit segment Slope is the tangent of angle we want to construct
QUICKER WAY: Cut the upper triangle in half ( 5:47) , you get a right triangle with Hyp=X, and Adjacent side of 0,5. cos(36) = Adj/Hyp Hyp = Adj/cos(36) X = 0,5/cos(36) X = 0,618 !
...construct a pentagon, center it at the origin, draw horizontal and vertical lines from the vertices, and play-play-play with 18 degrees, 72 degrees, etc., all year long!
I solved it by considering the equation x^5-1=0, then disassembling the equation and using the sum rule to put the output into another equation. After solving the equation that you got you should get the cosine of 18°. Then you simply plug it into the formula cos^2(x)+sin^2(x)=1 to get the solution.
You get minus psi because you solved X^2 + X - 1 = 0 ; if you do the change of variable y= - X, you get y^2 - y - 1 = 0 which is the characteristic on why you find psi and phi (characteristic equation of the fibonacci sequence, and of lots of stuff).
Length is actually always positive, even in complex fields, because it's a vectorial norm. If you do a bit of topology, you'll learn that length is actually DEFINED positive and so if it wasn't positive, then it wouldn't be a length.
when speaking on length of complex number you are talking about the norm, the norm of z=a+bi is defined to be this: ⁿ√(|a|ⁿ+|b|ⁿ), usually you will take 2 as "n". as you can see it is defined to be the root of positive number so it is always positive. just random fact to the equation, the only numbers apart from 2 that i seen being used are 1 and ∞ and they are used when you are talking about vectors or matrices, not complex numbers
negative distance is same as moving to opposite direction, which can be flipped to positive by reversing the angle of movement for same effect. while we could talk about negative length sides, it is much simple to use simpler numbers. but if we talk about imaginary distance it's whole another thing
bprp showed that sin(18) = cos(72) = 1/(2*phi). Using a 36/36/108 degrees isosceles triangle, you can do virtually the same construction to show that cos(36) = sin(54) = phi/2. (Or continue bprp's construction and drop a perpendicular to the left side of the main triangle.) According to Wikipedia and Wolfram, the 36/72/72 triangle is know as the "golden triangle" and the 36/36/108 is known as the "golden gnomon". These 2 triangles are referred to as Robinson triangles in the Wikipedia article on Penrose tiling.
Yeah !! Blackpenredpen, are these triangles linked somehow with self-similar Penrose tiles ? They were considered long time as a simple mathematical curiosity, until they became the core of the recent discovery of pseudocrystals.
One more way. Use a protractor, draw a triangle having 18,90,72°. Use a ruler. Measure sin 18°. By calculating the value of perpendicular upon hypotenuse.
This just shows how hopeless is transcendental trigonometry that we are happy as kids if we can solve one particular triangle using it and the result is still convoluted...
Fun fact, I discovered that ancient Mayan architecture used angles of 18 degrees a lot recently, because 360/20=18, and they used base 20 so you know, go figure that they would use 18 degrees a lot in their arcitecture. But could another reason for them using 18 degrees be because as we have just found, sin(18 deg)=1/(2*the golden ratio)? Is it possible that they were aware of the connection to the golden ratio all along?
At 3:00 minutes into the video you dropped a line that made a right angle to the opposite side side of the initial isosceles triangle. You should have cut the opposite side with a line equal to X drawn with a compass. That would have given you an isosceles triangle. with both sides equal to X.
So, consider a 40/70/70 triangle. Specifically, BAC = 40 degrees, ABC = ACB = 70 degrees. Set a point D on BC and draw the line BD such that BDA is 30 degrees and BDC is 40 degrees. The lower half is an isosceles triangle. The upper half has a 30 degree angle, allowing you to use the sine rule. Set cos(40) to be x, and remember that sin^2 + cos^2 = 1. In the end, I have a cubic polynomial. Is there a better way to approach this?
Thanks for evaluating sin(18°) geometrically.Otherwise, it can be also find easily by using Trigonometry. But the question arises that how the Right Triangle 36°_72°_72° can be drawn. To construct the above triangle, "Divide a straight line segment in Medial Section".
Let A = 18° Therefore, 5A = 90° ⇒ 2A + 3A = 90˚ ⇒ 2θ = 90˚ - 3A Taking sine on both sides, we get sin 2A = sin (90˚ - 3A) = cos 3A ⇒ 2 sin A cos A = 4 cos^3 A - 3 cos A ⇒ 2 sin A cos A - 4 cos^3A + 3 cos A = 0 ⇒ cos A (2 sin A - 4 cos^2 A + 3) = 0 Dividing both sides by cos A = cos 18˚ ≠ 0, we get ⇒ 2 sin θ - 4 (1 - sin^2 A) + 3 = 0 ⇒ 4 sin^2 A + 2 sin A - 1 = 0, which is a quadratic in form of sinA Solve for A Since sin18 lies in 1st quadrant. Take positive value. 😗
![Blox Fruits Dragon Rework Update [Full Stream]](http://i.ytimg.com/vi/EqDAp8udhm0/mqdefault.jpg)
sin(18 degrees) with the quadratic formula: 👉 ruclips.net/video/p2k756gbwik/видео.html
If 18 degrees seems random to you, just remember that in radians it's pi/10
Just remember its 180/10 in degrees
tau/20
david plotnik Just remember it is 20 grad
20 Grads*
oh I'm stupid, sorry, I'll edit
For your next video you should find cosine of 72 degrees :)
cos(72 deg)=sin(18 deg) remember the cofunction identity cos(90-theta)=sin(theta)
thanks for sharing
Any time :-)
It probably does in some way because that's just how math works.
it was a joke...
In math, whenever we draw badly, we just say "just believe in the math" and everything is ok. 🤣
or the more formal way used in textbooks: “figure not to scale”
This reminds me of the time my math teacher was teaching us some graphing function and the line he drew barely missed a point so he just made the point bigger so it connected
As one of my old maths teachers used to say: "If I draw a square and tell you it's a circle, you treat it as a circle and that's that!"
Thanks!
It has been hard enough for me to accept the fact that the limit of the ratio, of any 2 sequential terms in the Fibonacci series, equaled The Golden Ratio. Now, I find the Golden Ratio is also 2 x sin (pi/10). This is almost as cool as Euler's equation. Many thanks!
"This is blue by the way"
If it was green it would die
Again, a great video. In high school, we learned the method of taking 5θ = 90° and doing tedious manipulations. This method is so much better and pure.
You're right! In India trying to solve a question in an innovative way is always not appreciated (especially by Math Teachers)
+blackpenredpen
Thanks for phi-guring it out.
;)
I will never forgive you for this *sin*
This is a golden comment.
Witnessed 3 puns in one comment
When self similarity is in the game SQRT(5) often comes over to hang out.
Inscribe a regular pentagon in a circle or radius 1, the vertexes are (1,0), (cos 72, sin 72),(cos 144, sin 144), (cos 72, - sin 72), (cos 144, - sin 144).
The average of these coordinates will be the center of the pentagon (0,0).
1 + 2 cos 72 + 2 cos 144 = 0
1 + 2 cos 72 + (2 cos^2 72 - 1)= 0
4 cos^2 72 + 2 cos 72-1 = 0
Apply the quadratic formula
cos 72 = -1/4 + sqrt(5) / 4
Dear Sir,
I really appreciated your efforts in this wonderful piece of geometry explanation. I really liked this new way of solving and obtaining the. Golden ratio...
Sir,. I discovered that 18°*5 = 90 degrees therefore I found the value of sin18° by algebra and so I wanted to share it with you
..
Let 18° =x
2x+3x =90°
2x= 90°- 3x
Taking the sine on both sides,
You obtain
Sin(2x)= sin (90- 3x)
2sinx cosx = cos 3x
Sir .. after solving these equations by eliminating cos(x) from both sides and converting cos^2x into sin^2x we notice that a quadratic equation is formed. We solve it and obtain the same value as you did using this amazing mind bending geometry , I.e( (5)^0.5 - 1)/4 .
You have to know, however, how to write cos(3x) differently. I checked that it's equal to cos^3(x) - 3sin^2(x)*cos(x), but that's not something I would know on top of my head.
Once you know that, however, it is really nice way!
Abdullah Kanee Thanks :) Yes, in my maths class in high school we didn't learn triple angles identities, and since then, I was mainly using numerical values for the angles (or sinx=x approximation ;) ). In general, sin(2x), cos(2x), as well as sin(x+y), sinx+siny etc. are probably more useful. But it's easy to forget that there are other possibilities, if you don't use them :D
"What a-cute triangle" "My mommy says I'm special"
You're a little angle
That was great. The base of the full triangle was 1/phi , that was very cool. Great to see the enthusiasm for math here.
Notice if u keep on cutting 72° u will get similar triangle over and over again...I would like to do infinitely many times...wow...!
You can use angle bisector theorem after bisecting 72°
That is
1/x=x/1-x
1-x=x²
X²+x-1=0
Give you x=-1+√5/2
Nice
As soon as I saw you making the 36-36 big triangle, I could smell golden rations coming up.
I figured it out when he showed the √5 in the quadratic formula.
72 18 90 right triangle is useful in regular pentagon construction
Hypotenuse of this triangle has length a lim_{n\to \infty} \frac{F_{n+1}}{F_{n}}
where F_{n} is nth Fibonacci number
The moment i get convinced you are the true math ninja...
U hit me with a blow so powerful that i realize how my previous notion was such an obscene understatement.
Thanks for the proof
And i'll do a copy pasta math joke
Q:Why don't you accept people to drink in your math party?
A:Cuz you cant drink and derive
(sorry :D)
This guy never stops to amaze me, isn't it?
I thought that (-1 + sqrt 5)/2 is the golden ratio but then I remembered that the golden ratio is made from the quadratic equation with negative b.
I have not been able to breath for several seconds.
That's an ALL RIGHT TRIANGLE
This is one of the very few youtube maths videos where I can honestly say; been there, done that got the (Fibonacci ) Tee shirt.
When I was 8 my teacher showed me a compass and straightedge construction of a regular pentagon in a given circle. 30 years later I was able to prove the construction using that triangle.
My DiffEq prof was just like this guy. Best math teacher I ever had.
There was a much faster way of getting to that quadratic equation, why didn't you do it this way?
the triangles are similar so corresponding sides are proportional
1/x=x/mystery number
cross multiply
x^2=mystery number
1=x+mystery number aka x^2
x^2+x-1=0
sigh...
Happy Friday,
It's over 100 deg F here....
Stay cool everyone!
btw I'm 12
I know u must be 12.
I'm very confused now, more than i was before
Just bc u r 12.
Stay cool, kid.
It's really HOT here... I am going to buy some cold drinks!
I would buy u one if u live in Los Angeles as well.
Can you tell me why I am having a feeling of a pentagon and golden ratio
36, 72 and 108 are the angles that appear in a regular pentagon, In fact the ratio diagonal:side is phi
Here is another interesting fact. If you draw a regular Pentagon and join all the diagonals, you get a smaller Pentagon inside. The ratio of the side of the bigger Pentagon to the side of the smaller Pentagon is golden ratio^2.
Also sine(666) = -(golden ratio)/2
#sacredgeometry
Vivek Venkatesan
Oh sh*t! That's f*kin deamon math!
Cus pentagon side×gold ratio=diagonal
Simply a perpendicular bisector. That is a simple ratio given by Euclid in Data as well as elements. It is also taught in Geometry. As well as a modified G-conjecture.
This is a very quick result but the method of constructing a rectangle from two differently sized 45° right triangles, one 30-60-90° right triangle and the 18-72-90° (rt. ) triangle is elegant & beautiful
4:06, at this point you could say that the second triangle has a missing side of x^2 since the triangle was changed by a factor of x from the first one. Then at 6:03 you notice it's the same as 1-x. Then your equation becomes clear that x^2 = x-1 and get your solution
You should have given a different way for getting to sin(18⁰)
Let A= 18
5A=90⁰
2A = 90⁰ - 3A
And then taking sine on both sides and solving
The most interesting part of this is that the triangle can have a negative and positive answer and how to understand it and not discard it ,
1years ago, i saw this.
Yesterday, one school math test answer was cos72.
But i used this, and prove what is cos72.
Thank you for many information.
Belief in the math is now my mantra for life
Hey look at that, that's pretty cool, golden ratio value is (1+sqrt(5))/2, the negative root is just the negative of that value.
I just saw the end of the video, I'm feeling kinda redundant lol
This is equal to (1/2) * phi^(-1). The cosecant is equal to 2 * the golden ratio. In the original 36-72-72 triangle the ratio of the sides is exactly the golden ratio.
This is not so surprising because the golden ratio is (1 + sqrt(5))/2 and all the angles of the triangle are measured in fifths of 180 degrees (for the isosceles one).
Actually there is big Phi and small phi so it might be even closer than you think. Your explaination though doesn't make immediately sense to me but I'm sure if I thought a bit about it, it would!
+Marcus I like to call them major and minor golden ratio.
Minor golden ratio is (1-sqrt(5))/2
so (sqrt(5)-1)/2 = -(1/2) * (minor golden ratio)
Ok. I belive it is not trivial that (sqrt(5)-1)/2 = -(1/2) * (1-sqrt(5))/2 or true. Minor mistake perhaps have been added for this comment. But what is also not trivial and that I believe to be even more impressive as far as I understand, major phi(golden ratio) equals 1 over minor phi(other golden ratio) and also 1 plus minor phi(other golden ratio).
minor phi = -1/(major phi)
It actually doesn't equal 1 + minor phi
I would love to see more videos that deal with complex numbers
Understand well with what you explain, the sign + on sin 18° because it is on first quadrant
You could use the compound angle formula to get the value of sin 15 = sin(45-30) = sin45cos30 -cos45sin30
The 72 degree triangle is also something else... it's a truncation of one arm of a pentagram (which is also closely related to the golden ratio).
We easily construct angle if tangent can be expressed with four arithmetic operations and taking square roots
Addition and subtraction can be realized by moving segments with compass , multiplication and division can be realised with Thales' theorem square root we will get after geometric mean with unit segment
Slope is the tangent of angle we want to construct
08:10, yep I can see (suspect) golden ration when I see sqr(5)/2 not mentioning plus or minus 2.
QUICKER WAY: Cut the upper triangle in half ( 5:47) , you get a right triangle with Hyp=X, and Adjacent side of 0,5.
cos(36) = Adj/Hyp Hyp = Adj/cos(36) X = 0,5/cos(36) X = 0,618 !
isn't that kind of cheating? :(
I don't know if there is such thing as cheating in Mathematics haha :p @@agfd5659
@@iamyou5299 I mean it as in, how do you know what cos(36°) is equal to?
@@agfd5659 ah.. OK maybe :(
...construct a pentagon, center it at the origin, draw horizontal and vertical lines from the vertices, and play-play-play with 18 degrees, 72 degrees, etc., all year long!
5:20 I couldnt watch anymore it was weird and hard to understand with blue very confusing
Tomas Molina oops.... sorry.
:)
My thoughts exactly. This channel is mathematical heresy
John Hare whats so hard about understanding that two sides of the triangle are equal?
It was a joke...
@@nacargod5110 really? I don't get it
Best maths channel on RUclips.
Could you do a video on the super golden ratio? 👀
Never knew the Ood liked math.
By the way sine of 18 degrees is also the secant of 36 degrees divided by four.
Also, the golden ratio is 2 multiplied by the cosine of 36 degrees
I solved it by considering the equation x^5-1=0, then disassembling the equation and using the sum rule to put the output into another equation. After solving the equation that you got you should get the cosine of 18°. Then you simply plug it into the formula cos^2(x)+sin^2(x)=1 to get the solution.
You get minus psi because you solved X^2 + X - 1 = 0 ; if you do the change of variable y= - X, you get y^2 - y - 1 = 0 which is the characteristic on why you find psi and phi (characteristic equation of the fibonacci sequence, and of lots of stuff).
The final answer is equal to 1/(2phi), which is the same as (phi-1)/2, phi being the golden ratio. :)
Excellent! You don't know how great it is! Thanks.
what if negative length exists in imaginary fields
Length is actually always positive, even in complex fields, because it's a vectorial norm.
If you do a bit of topology, you'll learn that length is actually DEFINED positive and so if it wasn't positive, then it wouldn't be a length.
There's always pseudo-Euclidean space...
when speaking on length of complex number you are talking about the norm, the norm of z=a+bi is defined to be this: ⁿ√(|a|ⁿ+|b|ⁿ), usually you will take 2 as "n". as you can see it is defined to be the root of positive number so it is always positive.
just random fact to the equation, the only numbers apart from 2 that i seen being used are 1 and ∞ and they are used when you are talking about vectors or matrices, not complex numbers
negative distance is same as moving to opposite direction, which can be flipped to positive by reversing the angle of movement for same effect. while we could talk about negative length sides, it is much simple to use simpler numbers. but if we talk about imaginary distance it's whole another thing
Dat Epic Fish no. If there's a direction, it's a vector. And the length of a vector is still defined as a positive number
at 10:29 u u were probably wrong, because half of -1+root5/2 = -1+root5 ( not -1+root5/4 )
Ummm no? Lmao
1/2 times (-1 + root5/2) = -1 + root 5/4
(-1 + root5/2) *Divided* by 1/2 = -1 + root5
Let x=18
5x=90
3x+2x=90
3x=90-2x
Apply cosine
cos(3x)=sin(2x)
4cos^3(x)-3cos(x)=2sin(x)cos(x)
Canceling cos(x)
4cos^2(x)-3=2sin(x)
Now use cos^2(x)=1-sin^2(x)
4(1-sin^2(x))-3=2sin(x)
Consider sin(x)=y
4(1-y^2)-3=y
4-4y^2-3=y
1-4y^2=y
And solve...!!!
amazing :D
bprp showed that sin(18) = cos(72) = 1/(2*phi). Using a 36/36/108 degrees isosceles triangle, you can do virtually the same construction to show that cos(36) = sin(54) = phi/2. (Or continue bprp's construction and drop a perpendicular to the left side of the main triangle.)
According to Wikipedia and Wolfram, the 36/72/72 triangle is know as the "golden triangle" and the 36/36/108 is known as the "golden gnomon". These 2 triangles are referred to as Robinson triangles in the Wikipedia article on Penrose tiling.
5:00 the third line of the red is x², right? Similar triangles rule
And since it could alternatively be 1-x as you said 5:30, 1-x=x² familiar golden ratio
Lines aren't surfaces
Where's my PENTAGON?
Yeah !!
Blackpenredpen, are these triangles linked somehow with self-similar Penrose tiles ? They were considered long time as a simple mathematical curiosity, until they became the core of the recent discovery of pseudocrystals.
One more way. Use a protractor, draw a triangle having 18,90,72°. Use a ruler. Measure sin 18°. By calculating the value of perpendicular upon hypotenuse.
This just shows how hopeless is transcendental trigonometry that we are happy as kids if we can solve one particular triangle using it and the result is still convoluted...
Fun fact, I discovered that ancient Mayan architecture used angles of 18 degrees a lot recently, because 360/20=18, and they used base 20 so you know, go figure that they would use 18 degrees a lot in their arcitecture. But could another reason for them using 18 degrees be because as we have just found, sin(18 deg)=1/(2*the golden ratio)? Is it possible that they were aware of the connection to the golden ratio all along?
my math teacher would kill me if i give different angles same arc
Let us take a moment to appreciate how he switches the markers so fast.
thanks!
you should work out the cosine too which is the vertical leg of the triangle: sqrt(1-(-1+sqrt(5))/4)²)
At 3:00 minutes into the video you dropped a line that made a right angle to the opposite side side of the initial isosceles triangle.
You should have cut the opposite side with a line equal to X drawn with a compass. That would have given you an isosceles triangle. with both sides equal to X.
This is also why the golden ratio comes up when working with the pentagon.
I love your videos so much!! Keep it up mate!
sqrt(2-2cos(36)) = (-1 + sqrt(5))/2
Maths is so cool
Unbelievable
Why don't we learn this at schools or college 😟
We did. Did you never take trigonometry in high school?
Can you please explain the golden ratio part?
U make me recall the enjoyment that I got when I took up maths
Thanks to geometry problems like this I love abstract subjects in college, however in probability or physics I get lost.
GOOOOLLLLLDEEEEE NNNNN NNN RRRAAA TTTT IIIIIEEIEIIEIIII OOOOOOOO!!!!!!
So, consider a 40/70/70 triangle. Specifically, BAC = 40 degrees, ABC = ACB = 70 degrees. Set a point D on BC and draw the line BD such that BDA is 30 degrees and BDC is 40 degrees.
The lower half is an isosceles triangle. The upper half has a 30 degree angle, allowing you to use the sine rule. Set cos(40) to be x, and remember that sin^2 + cos^2 = 1.
In the end, I have a cubic polynomial. Is there a better way to approach this?
I love these videos! I am hooked!
The answer is essentialy 1 / (2phi).
yes!
I have a black
I have a pen
ahh, blackpen
I have a red
I have a pen
ahh, redpen
Blackpen, redpen
ahhh
blackpenredpen
You made me witness this meme with my own eyes, how dare you
Miguel Bruzual this is soo old, it makes me cry of nostalgia
Blackpenredpenbluepen pen
BPRP.
Instructions unclear. Pen is stuck in pineapple.
That was fun to watch, thank you
Very cool, it was funny seeing you couldnt wait to tell us.
New table value!!!
Quick and simple
x=√(1^2+1^2-2*1*1*cos(36°))
Hey man, write a book! :D
Why didn't you figure out X with the sine rule? x/sin36=1/sin72, cross multiple, xsin72=sin36, x=sin36/sin72=0.618... (same as (-1+root5)/4),
I have another video on that too! : )
to find your X value in the triangle couldnt u use A^2 = C^2 +b^2 - 2bc * Cos A.
Much simpler method
1:27 Oh this video is all about the golden ratio
6:27 He is not talking about golden ratio! I guess I won!
11:27 I LOST
Wouldn't it be easier to just use cosine theorem to calculate x?
Then you need to know what cos(36) equals to, but you don't.
Ivan Petrov what about law of sines then?
@@ajayjoel then, you don't know what's sin (72)
After hearing the first chord, I thought I was watching mathologer...
No need to solve x^2 + x - 1 = 0. Divided by x it is a 1 + x = 1/x - the golden ration equation. Awesome video, BTW!
Thanks for evaluating sin(18°) geometrically.Otherwise, it can be also find easily by using Trigonometry. But the question arises that how the Right Triangle 36°_72°_72° can be drawn. To construct the above triangle, "Divide a straight line segment in Medial Section".
1/(2.sin(18)) = 1/(2.((−1+√(5))/4)) =1.61803
I love when u say believe in the math
Somebody should make a shirt out of this:
"BELIEVE IN THE MATH"
Sin 18° = (√5-1)/4
A=18
2A=36
3A=54
5A=90
Sin (5A-3A) = Sin (2A)
Now, Sin(90-x)=Cos(x)
Cos(3A) = Sin (2A)
Solving this real quick by using identities......
4Cos³(A) - 3Cos(A) = 2Sin(A)Cos(A)
4Cos²(A) - 3- 2Sin(A) = 0
4-4Sin²(A) - 3 - 2Sin(A) =0
Sin(A) = x
Solve Quadratic
Sin(A) = (√5-1)/4
So, Sin(18°) = (√5-1)/4
Este video ha estado muy entretenido, Te felicito, Ahora ya puedo relacionar 18 con la proporción áurea.
With sen 18, we can have sen 64 which is sen 2^6, so we can get sen 1 :)
Let A = 18°
Therefore, 5A = 90°
⇒ 2A + 3A = 90˚
⇒ 2θ = 90˚ - 3A
Taking sine on both sides, we get
sin 2A = sin (90˚ - 3A) = cos 3A
⇒ 2 sin A cos A = 4 cos^3 A - 3 cos A
⇒ 2 sin A cos A - 4 cos^3A + 3 cos A = 0
⇒ cos A (2 sin A - 4 cos^2 A + 3) = 0
Dividing both sides by cos A = cos 18˚ ≠ 0, we get
⇒ 2 sin θ - 4 (1 - sin^2 A) + 3 = 0
⇒ 4 sin^2 A + 2 sin A - 1 = 0, which is a quadratic in form of sinA
Solve for A
Since sin18 lies in 1st quadrant. Take positive value.
😗
I'm amused by how your profile picture looks like a ninja when it's smaller.
This video is gold!