13:00 for that we can use the polar form of each numbers. For the first, we get e^(2*i*pi/3), then we divide by 3 the exponent (because of the cube root). So at the end we get 1/2(e^(2i*pi/9)+e^(-2*i*pi/9)) Then we can use the trig form and we see that the imaginary parts cancels: (cos(2pi/9)+isin(2pi/9)+cos(2pi/9)-isin(2pi/9))=2cos(2pi/9) Then we multiply by 1/2 and we get cos(2pi/9)≈0.766 In degrees, it gives us cos(40°) which is indeed sin(50°) using trig identity, like in the video

Man, the people who discovered this stuff without the use of calculators to sanity check what they were doing... hats off.

Finally, we can write out the value in roots of sin(1°), which we could then use to get the sine of any integer (despite lengthy expansions.

I really enjoy when you keep things real and show the struggles you had to reach the result and not only the cleaned-up result. As a math teacher I relate and get a better appreciation of your video. 谢谢

I’m finally far enough in my trig that, I was actually able to jump ahead of blackpen and derive the equation myself and then check my solution, just needed a push in the right direction. Well done Blackpen🤝

I've always had troubles with the cubic formula, and I believe it's all because weird stuff goes on with branch cuts when you have to take a sum of two nonreal cube roots. You'd have to go back to the derivation of the formula, and make sure everything is defined over a branch cut C^3 -> C so that you make sure you equate the compatible cube roots. It all feels very messy and fiddly and I don't want to be the one who has to delve into it.

always knew a video like this was coming

Excellent. Thanks very much ! 🙂

You can reach a formula that always works this way: You can begin using Euler’s formula and say that: Sin(x/3)=(1/2i)(e^(ix/3)- e^(-ix/3)) Next, you can just say that e^(ix/3)= cuberoot( cosx + isinx), and the same can be done to e^(-ix/3). This gives us the formula: Sin(x/3)=(1/2i)(cuberoot( cosx + isinx)- cuberoot( cosx - isinx)) This formula only works for x less than pi but greater than 0. If you want a formula for x less than 2pi but greater than pi, you should multiply the first term in the formula by w1, and multiply the second term by w2, where w1, and w2, are the complex cubic roots of unity. So we get the formula: Sin(x/3)=(1/2i)((-1/2+√3/2 i)cuberoot( cosx + isinx)- (-1/2-√3/2 i)cuberoot( cosx - isinx)) If you want me to explain why these formulas always work, feel free to reach out to me as it would be hard to explain here in the comment section. It has something to do with the rotations in the complex plane.

Good writing sir❤❤❤

Well, I love this I think your next video is to finally get the exact value of sin1° that is mixed with pythagorean theorem, your special special right triangle of 15-75-90, 18-72-90, 3-87-90 Then you will prove cardano's cubic formula, the trigonometric identities and more

In Cardano's formula if complex number show up it is not possible to get an answer than involves only reals and radicals.

Interesting results :)

We could have just used euler form and demoivre's theorem

A basic difficulty I have about this is: y=x^3+px+q has constant p and q. But here we have q=q(x). Typically we're trying to find the roots of a fixed polynomial. We vary the x and get the y value along a specific curve, and we're looking for where y=0. Here, as we vary x, the whole curve changes because q=q(x). Or in terms of 2nd order poly.: y=Ax^2+Bx+C(x). We could just use the quadratic formula, and get a result, but thinking about what exactly we are doing is a bit confusing. Could even consider 1st order: y=ax+b. Root formula would be x=-b/a. Now we try it with y=ax+b(x), and use the same formula, so x= -b(x)/a. I'll have to think about this a bit more. Any comments? Thanks

I’m not sure how correct I am, but i read on a wikipedia page that unfortunately according to Galois theory, for cubic equations with all 3 roots being real irrational numbers, the roots cannot be expressed with a finite radical expression using only real numbers. So unfortunately, we can’t write sin(10 degree) in a finite radical expression without the i. If someone could fact check me, that would bemuch appreciated

Congrats on 1000000ln3 subs🎉🎉🎉

isnt sin(10) an exact value, what makes a bunch of roots more exact than sin(10), given those roots are also just operators like sin()

I literally plug sin(x/3) in the calculator earlier This help me a lot

This is me every other class but math 😂

Those two complex numbers you wrote in your expression for sin 10° have magnitude 1, so you might try writing them in polar form and dividing their arguments by 3.

He just weaves in all conpcept in so easily It seems every sentence is just a revision of a topic

Keep it up bro, What's the best books to learn advanced mathematics after graduating high school?

13:00 you can set the expression in parenthesis to x and then ''cube both sides'' similiar to this vid: kzfaq.info/get/bejne/a9aRZKSj3t_QoIE.html You can then substitute x to get a cubic equation with real coefficients. However it seems like it will have 3 roots, probably because we cubed both sides. Which root is the correct answer? I dunno

Since the roots of a polynomial are not sorted and are algebraicaly indistinguishable, I claim there is no way to know what is the correct root for a given angle in advance. But you can have a set of solutions called red, blue, and black, which is kind of better.

bro really cube rooted the cube root of unity at some point

This is very similar to a Hong Kong 2007 grade 13 A-level mathematics problem. I regret I had arithmetic panic that time and I didn't score this. The whole paper is damn easy

Do you go from sin(50°) to sin(-70°) and then to sin(-190°)=sin(10°) by utilizing the cube roots of 1 because those cube roots are 120° apart?

I was just about to tell you to use cube root of unity, but then you figured it out yourself

This is nice,but at the end what you get is cos80=sin10 .the formula you get at the end after simlifying is :0.5(e^(4pi/9)i+e^-(4pi/9)i) which is cos(4pi/9)=cos80=sin10,so there is nothing special here,but the way you got it was nice

Interesting and very well done. Thank you for this!

11:16 you could prob put the terms inside the radicals in terms of cis to simplify the radicals

What is the alternative solution of the strend problem solved by ramanujan?

I love math.Thank you sir... ❤❤

Hi Blackpenredpen. Make a video demonstrating why π is irrational. I am subscribed to your channel, very good videos. Greetings from Mexico

this is a lot of formulae . It's better if you use euler's formula and some rearangements ; like expressing sine of x|3 as -i ×half of e^ix|3 - e^ix|3 then writing i as e^pi|2, -i as e^-pi|2. This reduces to your obtained answer.

Math is so beautiful and helpful in our life.❤️❤️❤️

how big is your board?

This man is mathing which we cant even math about mathematics

Blackpenredpen: I don't know what the exact value That Euler's formula: Hold my beer

His name should be written Guinness world record for the fastest pen changing

I have been watching your videos since I was about 13. Now I am 16 and I can gladly say that I can understand almost all of your videos. (Some of them still confuse me, but just wait another year!!!)

Another method to evaluate the complex sued is by assuming it to be u Then u^3 = -1 + 3(cbrt(-1/2 + iroot 3/2)(cbrt(-1/2 -root3/2)u U^3 = -1 + 3(1)u So this becomes a polynomial that is p(x) = u^3 -3u +1 Then again applying cardanos formula for depressed cubics you can get an answer @blackpenredpen I just checked with wolfram alpha I got an 3 real roots and when you divide one of the answers namely 0.3470 by 2 in the original formula you get 0.1735 which is very close to sin 10 degrees

I can just use the power series of sin to get a decimal approximation. I was hoping this would end with some sort of real closed-form radical expression. (knowing my luck, if i tried to manipulate the imaginaries in the radicals with polar forms and such, i'd end up with sin (10) = sin (10) )

Very interesting. I think one problem could be that at around 9:40 you say sqrt(-cos^2 x) is i*cos x. That's not correct; you are missing abs(...). sqrt(-cos^2 x) is +-i*abs(cos x). In the situation you are calculating that it makes no difference because there is +- in front. However, later at 10:38 it makes a difference.

I was wondering if you could do this problem sometime. This is a very interesting problem, and I can't get the proof anywhere. Prove that a^x = log (base a) x (a

You can use de movies theorem to find the formula for sin(x/3) much faster, and for the final bit you can use polar exponent form. I know you didn’t want to go into complex too much because that restricts the viewers who understand it and you’d have to teach them an entire new subject which is fair.

Interesting that this is an algebraic number. Can we say when this is the case and when sin(x) is transcendental?

once you have the sinx+icosx under roots, can you use euler identity to simplfy it further?

14:00 it is because you forgot the 1/64 perviously. + to check it the number is real simply check if it is equals to its cojuge aka replacement every "i" with "-i"

being a cube root it has 3 solutions for each cubic where 2 of the solutions are sin(10) = 1/2 [ cos(80) + i sin(80) + cos(80) + i sin(-80) ] which simplifies to cos(80) and cos (-80) so it is a long way of proving sin(10) = cos(90 - 10) and sin(10) = cos(10 - 90) surprising the other solutions that produce real solution are cos (40), cos(-40), cos(160), cos(-160)

4:54 “then we know that” me: what kind of ultra specific formula is that?

Is there a discord group or something where we can ask questions. I got a hard one lol.

8x³-6x+1=0 Where x=sin(10) This equation also have the same three roots We can also get approximate value by (10×π)÷180

Nice

This works from 0 to π: sqrt(1 - (1/2 (1/(cos(x) + sqrt(-1 + cos^2(x)))^(1/3) + (cos(x) + sqrt(-1 + cos^2(x)))^(1/3)))^2) And this works from π/2 to 2π + π/2: 1/(2 (-sin(x) + sqrt(-cos^2(x)))^(1/3)) + 1/2 (-sin(x) + sqrt(-cos^2(x)))^(1/3)

While trying to simplify the complex solution. I got cos(2pi/9) in the way and it is interesting that it is equal to exactly sin50 degrees

If you want to take the cube root of those imaginary numbers, you’re going to have to use DeMoivre’s Theorem. It states the following: For any number a+bi, it can be represented in the form r(cos(a) + isin(a)) or rcis(a). Additionally, for any complex number in this form that has a power applied to it, (r*cis(a))^n = r^n * cis(n*a). For example, if we are to take the fourth root of 1, we know that 1 equals 1(cos(0) + i*sin(0)), or cis(0). Keeping in mind that we can rotate this angle 360 degrees, we can represent 1 as cis(0), cis(360), cis(720), and cis(1080) (you’ll see why we do this in a second). Because we take 1 to the one fourth, or 0.25th power, that means: 1^0.25 = (cis(0))^0.25 = cis(0*0.25) = cis(0) = 1 We can also repeat this for the other reference angles: (cis(360))^0.25 = cis(360*0.25) = cis(90) = i For cis(720), the same process results in cis(180), or -1, and for cis(1080), this results in cis(270), or -i. As such, we can get that the fourth roots of one are 1, i, -1, and -i. Hope this helps!

Thats fine

We know the expression is real without checking because it is of the form z + z* where z* is the complex conjugate of z. z + z* is guaranteed always real. Now about the nice expression involving only radicals, I have no fuckin idea 😅

w0 works for the range 90-180, w2 for the range 0-180. I have a code in R plotting all the options. I will post the code if there is interest.

Featuring sin(10°) 😂

Hi! I made up a function that is almost perfectly equal to e^(-x^2), hoping for that we'll be able to come up with an elementary integral function for it. The function is 9/(7+2,8^(2,8x)+2,8^(-2,8x)), I created it using the reciprocal of cosh(x). How should I go on to make it more (or perfectly) equal?

PROF U have potential of being Profi Kray teacher's international MATH competition gold medalist)

Sir, would you please find out an accurate way to trisect any angle?

Please could u do a video on why sin(54 degrees) = φ/2 , where φ is the golden ratio? I feel like this is a result not many people know about. Thank you!

Sorry for asking an unrelated question. There is a statement "For complex numbers, square roots can only accept principal values as output." Is that statement true? Example: √-1 = i, but not -i, while i² = (-i)² = -1 If yes, some of your previous video may have wrong answers!

can u answer this : prod n = 2 to 20 (n ^ 3 - 1)/(n ^ 3 + 1) I try but I don't know how

And now this formula found...

Still waiting for that sine of 1 degree.

Revolutionary

always liked doing these things

9:57 why is the absolute value not needed?

This cubic equation has three roots, which would be [e^(i2π/9) + e^(-i2π/9)]/2, [e^(i8π/9) + e^(-i8π/9)]/2, and [e^(i14π/9) + e^(-i14π/9)]/2. The third of these roots gives cos(280°) = sin(10°) Before taking the cube roots we have e^[±i(2π/3 + 2πj)], j = 0, 1, 2. The three roots above come from e^[±i(2π/3 + 2πj)/3], j = 0, 1, 2.

"Can we tell which solution will give sin(10°)?" I may have an answer to your question. When solving for cos(θ/3) we have x_j = [cos(2πj/3 + θ/3) + i sin(2πj/3 + θ/3) + cos(-2πj/3 - θ/3) + i sin(-2πj/3 - θ/3)]/2, j = 0, 1, 2 When j = 0 we get x_0 = [cos(θ/3) + cos(-θ/3)]/2 = cos(θ/3). So, when solving for cos(θ/3) use the first root corresponding to j = 0. But, when solving for sin(θ/3) things are a bit more complicated... x_j = [cos(2πj/3 + γ/3) + i sin(2πj/3 + γ/3) + cos(-2πj/3 - γ/3) + i sin(-2πj/3 - γ/3)]/2, j = 0, 1, 2 where γ = θ + π/2. Then, after combining terms, substitute θ + π/2 back for γ and set j = 2 to get x_2 = [cos(3π/2 + θ/3) + cos(-3π/2 - θ/3)]/2 = sin(θ/3)! In summary when solving for cos(θ/3) set j = 0 and when solving for sin(θ/3) initially replace θ with γ - π/2 and set j = 2. Then replace γ back with θ + π/2 and simplify. So, similar to what you observed, solving for cos(θ/3) requires the first root (j = 0) and solving for sin(θ/3) requires the third root (j = 2).

It is impossible to write an expression for a non-multiple of 3 degrees for sine without having a cube root of a complex number (or worse). By the way, here are some expressions for sine of non-multiple of 3 degrees that I find interesting. Note that you can copy and paste them into Wolfram Alpha in order to visualize them (and verify that they work). You have to be careful when taking the cube root of a complex number below. Choose the "principal value" (since there are 3 possibilities, but only one fits the conventional definition for principal value). I avoid writing "i" to make the syntax more universal for computer entry. Also, sind() is the sine degrees function. First the simplified version of what was derived in the video: sind(10)= sqrt(2-(1/2+sqrt(-3)/2)^(1/3)-(1/2-sqrt(-3)/2)^(1/3))/2 Now here is an alternative form that only has one cube-root in it: sind(10)= 1/sqrt(1-(1-4/(2-( 4+sqrt(-48))^(1/3)))^2) And finally, the holy grail, for 1 degree: sind( 1)= 1/sqrt(1-(1-16/(8-(4+sqrt(-48))^(1/3)*(sqrt(-1)-sqrt(-5)+sqrt(10+sqrt(20)))))^2)

Ah yes, finnaly, a non ridicolous degree.

Just substitute. a fot the first radical and b for the second one. Then the product ab would be the cubic root of the product of conjugated which is real. Then a^3 +b^3 would be the sum of conjugates which is also real. Those two quantities will generate a cubic equation of a+b which has real roots. I suggest using a numerical method to solve as opposed to cubic formula.

On this regard I would like to recommend you to check out the book called "One-third Angle in Trigonometry: New method for Cubic Equation" by Bhava Nath Dahal, which is available online. I am to greedy to pay a quid for it, but as the description says, it might have the general formula we all are looking for. If someone does buy this book, please share the answer :)

Hi blackpenredpen! A challenge for you from my side - prove the following: The volume of an n-dimensional sphere of radius R is : (Pi to the power (n/2))/(n/2) factorial

You should have pointed out that sin 50 = sin 130 and -sin 70 = sin 250. The three values you calculated are for angles all 120 degrees apart. 10, 130 and 250 degrees.

Here is another challenge. 18° = 60° - 3*14°. Thus, given x = sin14° evaluate sin18° as a function of x. That solution is relatively simple, but let us go the other way around. Given that sin18° = (Φ - 1)/2 = 1/(2Φ) where Φ = (√5 + 1)/2 is the Golden Ratio, evaluate sin14° as a function of Φ, where 14° = (60° - 18°)/3. I'm more impressed by how fast @blackpenredpen switches between pens. I try that and there's more ink on me than the board.

Could someone explain to me why even though he plugged in x = 30° he still got the sin(150°)? Is there a complex analysis reason for this in understandable terms?

Very interesting. That polynomial has one real root and two imaginary roots which are of course conjugates.

Could you pelase help to integrate 1/(x^2+x+1)^2? It is a hard one. 🙂

-sinx+icosx=(-i)(cosx+isinx)=-ie^ix -sinx-icosx=(i)(cosx-isinx)=ie^-ix Cube root of i= -i, root(3)/2+/-(i/2) Cube root of -i= i, -root(3)/2+/-(i/2) Wait... I'm going to get back to the polar form expression for sin. This doesn't help you go further... but it does give you an alternate route to prove the same expression. Maybe exploring it will help you understand which form to use.

Apply the third using 60 and then halve it using that original one?

i love this

bring a video on bayes theorem

De Moivre's formula?

Can someone explain to me the cube roots of unity? Is it as simple as "assuming you have 1 answer, you can find the other 2?" .....assuming you only have 1 eeal root, if all 3 roots are real i doubt unity would work as youre introducing unessesary "i"s Anyway so like if I had a cubic equation whose first root is... idk 7, could I find the other two roots using omega? Or is there some sort of special requirements for this to work? (For example... idk maybe you require two terms like in the video)

What is the sine of an imaginary number? Or the cosine of such?

..working on it from time to time.. not there yet with a common usable formula ...so faar the function is off by +2.7 degres at 'sin(10)' degrees ..but its a start... ..if someone want to help pitch in this is the startinng formula, x is in rad, and the output is in sin of x +0 to 3 degrees of error, with an error at zero and pi/2 of zero (3 degees of error at sin(45) ) to far off for any use yet Y= (-pi(x)^2 + pi^2x) / ( 2pi^2- (4pi - 2pi/11) )

Exact expression for sin(10): Use: cos(80) = sin(10) Use: half angle formula for cos: cos(A/2) = +/- sqrt((1 + cos(A))/2). Set x = 2/A, multiply by 2, assume 0

To solve the equation 4(sinx)^3-3sinx+sin(3x)=0, substitute sinx=u+v to obtain u^3 = -1/8*sin(3x)±i/8*cos(3x). Before taking the cubic root to obtain u, we must multiply u^3 by e^(2𝜋i*n), n=0,1,2 to obtain all three branches of the cubic root. Then we have u = ( -1/8*sin(3x)±i/8*cos(3x))^(1/3)*e^(2𝜋i*n/3), n=0,1,2. Since sinx = u+v and v=1/(4u) we get sinx = (1/8*sin(3x)+i/8*cos(3x))^(1/3)*e^(2𝜋i*n/3) + (1/8*sin(3x)-i/8*cos(3x))^(1/3)*e^(-2𝜋i*n/3) , n=0,1,2. For 3x=𝜋/6 we get sin(𝜋/18) = ½*e^(2𝜋i/3)^(1/3)* e^(2𝜋i*n/3) + ½*e^(-2𝜋i/3)^(1/3)* e^(-2𝜋i*n/3) =cos(2𝜋/9*(1+3n)) = cos(40°)≈0.7660.. , -cos(20°)≈-0.9396.. , sin(10°)≈0.1736.. , for n=0,1,2, respectively. Obviously, we take n=2 to obtain the correct solution. /Roland Karlsson

my approach was to look at those two radicals in polar form, and realize that each has three roots in the complex plane, all equal in magnitude but splayed out 120 degrees from one another. the magnitude of the quantity in each radical is 1, so you just have to determine the three phase angles. for the first radical, the possible phase angles are 2pi/9, 8pi/9, and 14pi/9. (roughly 40, 160, and 280 degrees if you prefer thinking in degrees) for the second radical, the possible phase angles are 4pi/9, 10pi/9, and 16pi/9. (roughly 80, 200, and 320 degrees) If you draw these in the complex plane, you'll see that each triad is splayed out 120 degrees (2pi/3) about the origin. Then, you need to add these two values. Consider all possible combinations of the three, and figure out which ones have wholly real-valued sums. This is easy, just pick off the ones in "group 1" and "group 2" which have equal and opposite imaginary values. There are three possible additions which result in a real-valued result: (1) 2pi/9 from the first radical with 16pi/9 from the second radical. (2) 14pi/9 from the first radical with 4pi/9 from the second radical. (3) 8pi/9 from the first radical with 10pi/9 from the second radical. Each of these pairs has equal and opposite imaginary components, so all you have to do is add the real components. It turns out that in each pair, the real values are identical, so you pick one and double it. And then as the last step you multiply by 1/2 to complete the expression from the video. combination (1) gives you sin(50). (coming from sin(150) in the 1/3-angle formula) combination (2) gives you sin(10). (coming from sin(30) in the 1/3-angle formula) combination (3) gives you sin(70). (coming from sin(210) in the 1/3-angle formula) The problem with Wolfram is that it arbitrarily chooses one of the three roots and discards the others. Apparently, the roots chosen correspond to combination (1), which is why you got the result for 50 degrees. In this case, combination (2) is the one you want. (my final quandary was that there are FOUR angles which have sines of +-1/2: 30, 150, 210, and 330 degrees. The first three are represented in the combinations above, but I couldn't find a combination corresponding to 330 degrees. That, in turn, would give us sin(110) when put through the 1/3-angle formula. However, sin(110) = sin(70) so that simply collapses into combination (3) above). This continues: sin(390) degrees is also equal to 1/2, and plugging that into the 1/3-angle formula would give you sin(130), but that's the same as sin(50), so it collapses into combination (1). And sin(360+150) = sin(510) = 1/2, which would give sin(170) after dividing the angle by 3, but that's exactly the same as sin(10) which corresponds to combination (2). This continues forever....)

Hey, the values you get in first place and in the first check are sin(60-10°) and sin(60+10°). using sine addition formula for each and subtracting the equation you get sin(10°)=-(-sin(70°)+sin(50°))/(cos(pi/3)*2). Not the answer to why you choose a solution respect to the other but it was funny to note the simmetry around pi/3

Is it possible using only real numbers?

I think what you are really doing is getting sin(x/3) = Im(e^(xi/3) = [e^(xi/3) - e^(-xi/3)] / 2i . If you substitute for the exponentials using Euler's formula, you will end up with the same 1/2(sum of cube roots) that you got after simplifying. But I don't think this helps to find a closed form using real numbers?

The final solutions can be changed into hyperbolic functions sinh (ix) = i sin (x), cosh (ix) = cos (x) to verify your answer , probably without the use of calculator.