this special triangle gives us sin(18º) 

sin(18 degrees) with the quadratic formula: 👉 ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-p2k756gbwik.html

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

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

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

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!

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

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

"This is blue by the way"

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

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.

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

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.

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 .

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...!

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

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.

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.

I love these videos! I am hooked!

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

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

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

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.

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

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

Hey man, write a book! :D

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

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

That was fun to watch, thank you

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

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

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

This video is gold!

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

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

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 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

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

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

Subscribed!

Belief in the math is now my mantra for life

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)

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?

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

This was cool!!

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

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

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 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).

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

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.

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.

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

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

Never gonna forget this now !!

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)

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

Would it be possible to solve using "a" instead of 1 for the triangle sides?

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).

Best maths channel on RU-vid.

thank youuuu so much for this amazing video

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

Thanks!

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

Can you please explain the golden ratio part?

MINDBLOWING VIDEO, SO AWESOME

I love when u say believe in the math

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 ,

Very nice. I wonder now is there an example of an Cubic of the casus iiriducible ( three real zero s ) that are in the first place expressed in sinus or cosus and then in this way can be expressed in roots anyway?

I saw what you did there. Awesome!

blew my mind!!!

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 ?

awesome vid good !

Pretty cool!

very awesome!

That was so beautiful, I thought I'd die!!!

FYI, you used masculine ordinal indicator instead of degree sign. It looks strange with some fonts. (Like in the page title on my browser's tab.)

Awesome videos

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

"This is sooo coool" A very special channel in RU-vid!

Never knew the Ood liked math.

I really enjoyed

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

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.

this is amazing

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

You guessed I would comment about the golden ratio but I know sin and pi are linked so what is the link between pi 18 and the golden ratio

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

Excellent video

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

Thanks very much

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...

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).

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.

...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!

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.

unbelievable!!! incredible!!