sin(18 degrees) with the quadratic formula: 👉 kzfaq.info/get/bejne/ppibaphnzMjbmZ8.html

If 18 degrees seems random to you, just remember that in radians it's pi/10

In math, whenever we draw badly, we just say "just believe in the math" and everything is ok. 🤣

For your next video you should find cosine of 72 degrees :)

"This is blue by the way"

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!

When self similarity is in the game SQRT(5) often comes over to hang out.

+blackpenredpen Thanks for phi-guring it out. ;)

"What a-cute triangle" "My mommy says I'm special"

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...!

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

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.

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

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.

That was great. The base of the full triangle was 1/phi , that was very cool. Great to see the enthusiasm for math here.

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 .

Can you tell me why I am having a feeling of a pentagon and golden ratio

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.

08:10, yep I can see (suspect) golden ration when I see sqr(5)/2 not mentioning plus or minus 2.

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.

As soon as I saw you making the 36-36 big triangle, I could smell golden rations coming up.

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.

5:20 I couldnt watch anymore it was weird and hard to understand with blue very confusing

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

I would love to see more videos that deal with complex numbers

This guy never stops to amaze me, isn't it?

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

GOOOOLLLLLDEEEEE NNNNN NNN RRRAAA TTTT IIIIIEEIEIIEIIII OOOOOOOO!!!!!!

I have not been able to breath for several seconds. That's an ALL RIGHT TRIANGLE

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

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

You could use the compound angle formula to get the value of sin 15 = sin(45-30) = sin45cos30 -cos45sin30

My DiffEq prof was just like this guy. Best math teacher I ever had.

Hey man, write a book! :D

I love your videos so much!! Keep it up mate!

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

Excellent! You don't know how great it is! Thanks.

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)

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 ,

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?

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.

I love these videos! I am hooked!

Este video ha estado muy entretenido, Te felicito, Ahora ya puedo relacionar 18 con la proporción áurea.

Understand well with what you explain, the sign + on sin 18° because it is on first quadrant

you should work out the cosine too which is the vertical leg of the triangle: sqrt(1-(-1+sqrt(5))/4)²)

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.

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).

what if negative length exists in imaginary fields

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.

5:00 the third line of the red is x², right? Similar triangles rule

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

Belief in the math is now my mantra for life

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.

Could you do a video on the super golden ratio? 👀

I have a question, how come at the beginning you couldn’t just use cosine rule to find the length of x? I tried it myself and got a different value

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).

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".

Very cool, it was funny seeing you couldnt wait to tell us.

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...

This is also why the golden ratio comes up when working with the pentagon.

The final answer is equal to 1/(2phi), which is the same as (phi-1)/2, phi being the golden ratio. :)

That was fun to watch, thank you

Hello Sir., May I Have A Question Regarding 18°, How do you simplify or Write it into Radian like This for Example "2π/4" ... Thank you & I Love your Videos... 😘

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).

Never knew the Ood liked math.

U make me recall the enjoyment that I got when I took up maths

`Hi, please do a video on a fourier transform :)

Buena pedagogia en este ejemplo: Una buena forma para ejercitar teorema de la bisectriz, clasificacion de triangulos, propiedades de angulos en triangulos y resolvente de la ecuación de segundo grado.

...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!

Can you please explain the golden ratio part?

So only with a 36-72-72 isosceles triangle does the line bisecting the 72 degrees equal the short side?

to find your X value in the triangle couldnt u use A^2 = C^2 +b^2 - 2bc * Cos A. Much simpler method

Could you please add the name of the piece of music which plays at the beginning of the video in the discription. The same goes for your other videos btw.

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.

Let us take a moment to appreciate how he switches the markers so fast.

Hello.How about making a video showing a method to calculate cubic root of any real number?

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.

I have a black I have a pen ahh, blackpen I have a red I have a pen ahh, redpen Blackpen, redpen ahhh blackpenredpen

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.

Best maths channel on KZfaq.

Somebody should make a shirt out of this: "BELIEVE IN THE MATH"

Couldn’t u also use law of cosines to find missing side then divide by 2 and then use Pythagorean thm for the height

Can you do this trick with other angle values???

I love when u say believe in the math

so with this method, you can find the sin etc. of any angle? even the 30, 45, 90?

sqrt(2-2cos(36)) = (-1 + sqrt(5))/2

Dear blackpenredpen sir! Thanks for the explanation! But I ve a doubt ! Which theorem do you use to get : x/1=1-x/x ? Or anyone else can explain me ?

Never gonna forget this now !!

Thanks to geometry problems like this I love abstract subjects in college, however in probability or physics I get lost.

Wouldn't it be easier to just use cosine theorem to calculate x?

This video is gold!

Now I will remember the Golden Ratio all the time

New table value!!!

I was think that you could just use law of sines to get x, but when I think abt it, that could’ve made it harder, idk fs tho and don’t feel like carrying out the math for it

"This is sooo coool" A very special channel in KZfaq!

as soon as I saw x^2+x-1=0, I knew what was coming and it was beautiful

why if I look for the value of cos 18° using the triangle the result is different from the original value of cos 18°?

I knew where this was going the instant you said “36 degrees”.

You definitely could've used the sin law to find x then divide by 2 (inexact however) or manipulate 18 degrees into the sum/difference of some known angles or even demoirve's theorem. With all that said, I appreciate your geometric illustration lol.