To be perfectly honest, though, how often in your life are you going to need to calculate a question like this without a calculator? Practically every smart device has some form of calculator in it nowadays d:
We have √(2+√3) = 2√((1+√3/2)/2) = 2√((1+cos(𝜋/6))/2) = 2cos(𝜋/12) Next, √(2-√(2+√3)) = √(2-2cos(𝜋/12)) = 2√((1-cos(𝜋/12))/2) = 2sin(𝜋/24). Knowing that for all x > 0, sin(x) < x, we have sin(𝜋/24) < 𝜋/24, so 2sin(𝜋/24) < 𝜋/12 meaning √(2-√(2+√3)) < 𝜋/12
Well, that's lame, realizing that it represented the relationship between a chord and an arc! As any real man does, I chose to completely overlook that clever and insightful solution and instead brute force calculate the whole square root expression in my head, with up to 5 decimal places. I (eveeentually) came up with the answer that the square root expression was almost exactly 0.261 and that pi/12 was about 0.2618. The calculator then confirmed this was indeed correct (0.26105 < 0.2618)! I was right, what a delight! 😄
P.S: I did actually make a slight mistake during one of the first steps, which led med to get the last step of calculating sqrt(0.06805) instead of sqrt(0.06815), but in the end, it didn't matter (I got the answer of 0.261^2 = 0.068121, which I concluded was close enough...)
@@michaelblankenau6598 Well, I think he said something in the line of not trying to estimate it... which I didn't, I calculated it. But the main point was, I would never have thought of the way he solved it. Also, I was actually kind of happy that I managed to calculate that in my head, so I felt I had some bragging rights... 😄
I think the key is analyzing that what makes you compare an expression with pi with an expression that has square roots, especially root 3. That leads naturally to trig, and clearly to start with an angle of pi/12 which leads to the 1/2 angle formula. But it is a contrived problem based on the trig and circle relationship and not something that would occur organically or as an outshoot of another exploration.
@@tamonekicofi For example of what you said, 24 sin(π/24) = 12 √(2 - √(2 + √3)) < π comes from a regular 24-gon since it has a central angle of 2 π and if we divide it into 24 triangles, the central angle of each is π/12. Bisect this and you get two congruent right triangles where the central angle is π/24 and we can apply trignometric definitions to find the ratios between the sides.
Constructing a perpendicular bisector of line c, passing through the center of the arc and also bisecting angle θ, we get the formula: c = 2 * sin(θ/2) For smaller and smaller angles of θ, the difference between θ and c will approach 0, and looking at the diagram it’s easy to see why.
I once wrote Theorem of Al-Kashi in a test. Teacher crossed it out and wrote Law of Cosines. I failed the test because of that. I'll never make that mistake again, it was a valuable lesson.
We learned this as as generalized Pythagorean theorem, also called as law of cosines. I really never heard about Al Kashi until now. This must be some new stuff made for some to feel better. Yes, many things today have theis names changed based on just feelings.
TEST TAKING HINT: The larger quantity is almost always the one with the pi. The question writers like these kinds of problems. The arc length will have a pi and the smaller segment will not. Just choose the pi and move on with the test. Come back to this one if there's time.
Depending on the test, you may have to show your work. But yeah, you'll usually still get something out of stating the correct answer, even if it's a wild guess.
not necessarily you could potentially divide by pi on both the quantities now the quantity that has pi in it is the one with the smaller value so it would be better if you had mentioned it works in most cases provided pi is in the numerator or so but anyways a good observation I will use that !!!
The shortest distance between 2 points is always a straight line (assuming we talking about conventional Euclidian geometry, of course). Since c is clearly a straight line (deliberately drawn that way) and theta is clearly not (as a part of circle), by definition c < theta.
Here is how I did it: Take a unit circle. Draw a sector with an angle of 30deg. Inscribe a triangle in that sector. The arc length of that sector is π/12. Calculate the length of the base (the side whose endpoints intersect the circumference) of that triangle using the cosine rule. It comes out to be sqrt(2 - sqrt (3)). Compare this expression with the one on the RHS in the question: on simple comparison, it can be observed that it is obviously bigger. However, since the arc of a circle is longer than a straight line for the same subtended angle, we can conclude that the LHS is bigger than the RHS.
To solve this you have to be intimately aware of standard theta value representations and cosines of pi. I am not, so I didn't recognize the initial relationship between the 2 samples given. I feel like this problem was kind of created through reverse engineering of sorts, haha.
Between this example then apply it to the interesting fact that the two towers on the Golden Gate Bridge are out of parallel by about 8" (if memory serves me correctly). This gives me the answer when a child asks that proverbial "Why do I need to learn this?" Thanks for sharing.
I had to use the brute force approach where i used the Pythagorean theorem in repeated application. First to compute the chord when it is 30 degrees , and then again for 15 degrees. Took me much longer than you guys but did not need any trig.
I considered one sector of a 24-gon and calculated from there. I had struggles with the square root though, as I kept using the different of angles law and not the double angle formula. So that was my mistake from there as I should have double angled pi/6 twice.
I looked at it and said the left > right, but I could be wrong. Then broke the rules and did a quick estimate in my head and concluded the same. I relized they would be close to the same value, but not as close as it turned out to be. I haven't done trigonomitry by hand in over 30 years, so that didn't occur to me.
Wow I solved it by guess work but not sure if that's correct.. π can be written as 22/7 then the whole LHS side can be written as 2× (11/(7×6) ) so on LHS we have a quantity multiplied by two and on rha we have square root of 2 minus square root of something.. that's obviously going to be smaller ...
My first thought was "Who cares?" The I realised that someone must care to have asked the question. I would assume that Pi/12 would still be a good sized portion, so I would go for that.
@@gavindeane3670 well, lets just take it numbers, without the square roots, since I don't know how to put them here. 2-(2+3) even if you do the squares, that's still 2-5... and since you go left to right, 2-5 is a negative number... square root of 2, minus the sum of the square root of 2 plus the square root of 3.... the part in the "nested" square is larger than the initial number... so... negative.
@@gwouru The part in the nested square root is not smaller than the initial number. There's no 2+3. It's 2+√3. √3 is about 1.732, so 2+√3 is less than 4, so √(2+√3) is less than 2, so 2 - √(2+√3) is positive.
@@gavindeane3670 you missed something... and I'm going to copy your symbol to help... √2 - (√2+√3) So... if we take it as is... with shortened numbers we get 1.414 - (1.414+1.732) so, that equals 1.414 - 3.146 = (-1.732) Or simplify negative √3
@@gwouru Are you watching the same video as me??? There's no √2 +√3. It's √(2+√3). And at no point are we ever taking the square root of 2 on its own, so there's no 1.414 anywhere. We have √(2 - √(2 + √3)) √3 is about 1.732 so 2 + √3 is 3.732. So we have √(2 - √(3.732)) √(3.732) is about 1.932, so we have √(2 - 1.932) 2 - 1.932 is a positive number.
an electric train goes 100 miles per hour traveling a 200 miles distance west and the wind is 20 miles per hour east. how fast the train's steam smoke coming out of the train steam stack and why?
he did note that the shortest distance between two points is a straight line. c is a straight line and the theta arc isn't. so that's not just an observation, just referring to a statement that doesn't really need proof (or if you want proof of that you can look it up).
@@user_uif_ghg_wer_das of course you can prove the shortest distance between two points on a flat surface is a straight line, calculus of variations is your friend there.
@@user_uif_ghg_wer_das Isn't there a formula for the circumference of polygons? Like an n-sided one has n times a formula like in this video. Then you could have a limit where n goes to infinity of this length steadily increasing and approaching 2π.
The right one is larger. I don't really know, but the confidence sells it. If it turns out to be right, I just sit back with the slightest of smiles. If it turns out I'm wrong, I just say, "Ohhh, you think the right is on THAT side.", and then they start questioning everything all over again.
aww, you had a 50% chance! my guess also, after I wasted some time figuring out (4+2√3) = (1+√3)^2 so the inner root can be simplified a little bit. and then I got stuck. so, you saved yourself some time, very efficient!
A good advanced example to learn Trigonometry.
To be perfectly honest, though, how often in your life are you going to need to calculate a question like this without a calculator? Practically every smart device has some form of calculator in it nowadays d:
You can do it knowing the method of computing pi with an infinite nested square roots from polygons approaching circles.
We have √(2+√3) = 2√((1+√3/2)/2) = 2√((1+cos(𝜋/6))/2) = 2cos(𝜋/12) Next, √(2-√(2+√3)) = √(2-2cos(𝜋/12)) = 2√((1-cos(𝜋/12))/2) = 2sin(𝜋/24). Knowing that for all x > 0, sin(x) < x, we have sin(𝜋/24) < 𝜋/24, so 2sin(𝜋/24) < 𝜋/12 meaning √(2-√(2+√3)) < 𝜋/12
I appreciate the great content. Sometimes I cannot easily follow it, but it makes me think out problems more.
If the difference is less than 0.001, then they are equal.
Sincerely,
A scientist
Pi = 3
_- Engineers_
@@angrytedtalks
pi=4
-politicians
Then what if it's 0.001 km ? Or even 0.001 light year ?
@@jud.7795 Arithmetic error. Same place.
@@jud.7795 still negligible. If not, use different units.
Well, that's lame, realizing that it represented the relationship between a chord and an arc! As any real man does, I chose to completely overlook that clever and insightful solution and instead brute force calculate the whole square root expression in my head, with up to 5 decimal places. I (eveeentually) came up with the answer that the square root expression was almost exactly 0.261 and that pi/12 was about 0.2618. The calculator then confirmed this was indeed correct (0.26105 < 0.2618)! I was right, what a delight! 😄
P.S: I did actually make a slight mistake during one of the first steps, which led med to get the last step of calculating sqrt(0.06805) instead of sqrt(0.06815), but in the end, it didn't matter (I got the answer of 0.261^2 = 0.068121, which I concluded was close enough...)
Nice memory
You did it the way he suggested not to do it . Which missed the whole point of the exercise.
@@michaelblankenau6598 Well, I think he said something in the line of not trying to estimate it... which I didn't, I calculated it. But the main point was, I would never have thought of the way he solved it.
Also, I was actually kind of happy that I managed to calculate that in my head, so I felt I had some bragging rights... 😄
I think the key is analyzing that what makes you compare an expression with pi with an expression that has square roots, especially root 3. That leads naturally to trig, and clearly to start with an angle of pi/12 which leads to the 1/2 angle formula. But it is a contrived problem based on the trig and circle relationship and not something that would occur organically or as an outshoot of another exploration.
It’s both beautiful and absurd.
It is literally part of early attempts to estimate Pi from placing regular polygons about and inside a circle because n > 2 implies n sin(π/n) < Pi < n tan(π/n)
And when you have only Euclidean geometric constructions to calculate with, all you may use are addition, subtraction, multiplication, division, and square roots.
6 sin(π/6) = 3 < π < 6 √1/√3 = 6 tan(π/6)
12 sin(π/12) = 6 √(2 - √3) < π < 12 √1/(2 + √3) = 12 tan(π/12)
24 sin(π/24) = 12 √(2 - √(2 + √3)) < π < 24 √(2 - √3)/(2 + √(2 + √3)) = 24 tan(π/24)
48 sin(π/48) = 24 √(2 - √(2 + √(2 + √3))) < π < 48 √(2 - √(2 + √3))/(2 + √(2 + √(2 + √3))) = 48 tan(π/48)
223/71 < 96 sin(π/96) = 48 √(2 - √(2 + √(2 + √(2 + √3)))) < π < 96 √(2 - √(2 + √(2 + √3)))/(2 + √(2 + √(2 + √(2 + √3)))) = 96 tan(π/96) < 22/7 [This calculation dates to Archimedes]
you can think of it as comparing circumference of a circle and a regular 24-gon inscribed into it
@@tamonekicofi For example of what you said, 24 sin(π/24) = 12 √(2 - √(2 + √3)) < π comes from a regular 24-gon since it has a central angle of 2 π and if we divide it into 24 triangles, the central angle of each is π/12. Bisect this and you get two congruent right triangles where the central angle is π/24 and we can apply trignometric definitions to find the ratios between the sides.
Constructing a perpendicular bisector of line c, passing through the center of the arc and also bisecting angle θ, we get the formula:
c = 2 * sin(θ/2)
For smaller and smaller angles of θ, the difference between θ and c will approach 0, and looking at the diagram it’s easy to see why.
Legends assuming π = 180° and saying π/12 is large 💀💀
I once wrote Theorem of Al-Kashi in a test.
Teacher crossed it out and wrote Law of Cosines.
I failed the test because of that.
I'll never make that mistake again, it was a valuable lesson.
We learned this as as generalized Pythagorean theorem, also called as law of cosines. I really never heard about Al Kashi until now. This must be some new stuff made for some to feel better. Yes, many things today have theis names changed based on just feelings.
You failed a test for a single wrong answer? Please...
Today, I learned that Theorem of Al-Kashi is the normal name as taught in France.
@@k1ry4n that is a wrong conclusion
@@engineboy_1449 Care to explain why?
TEST TAKING HINT: The larger quantity is almost always the one with the pi. The question writers like these kinds of problems. The arc length will have a pi and the smaller segment will not. Just choose the pi and move on with the test. Come back to this one if there's time.
Depending on the test, you may have to show your work. But yeah, you'll usually still get something out of stating the correct answer, even if it's a wild guess.
not necessarily you could potentially divide by pi on both the quantities now the quantity that has pi in it is the one with the smaller value so it would be better if you had mentioned it works in most cases provided pi is in the numerator or so but anyways a good observation I will use that !!!
Shouldn’t we more thoroughly prove that for all theta c
For all theta less than 2 pi you could say because a straight line is the shortest distance
The shortest distance between 2 points is always a straight line (assuming we talking about conventional Euclidian geometry, of course). Since c is clearly a straight line (deliberately drawn that way) and theta is clearly not (as a part of circle), by definition c < theta.
@@Ultramixing-p8e you didnt prove that the shortest distance between 2 points is always a line though.
@@jacobgoldman5780 under euclidean geometry the shortest distance being a straight line is an axiom/assumption so you do not prove it.
This seems like more of a mind-reading problem than a math problem
Here is how I did it:
Take a unit circle. Draw a sector with an angle of 30deg. Inscribe a triangle in that sector. The arc length of that sector is π/12. Calculate the length of the base (the side whose endpoints intersect the circumference) of that triangle using the cosine rule. It comes out to be sqrt(2 - sqrt (3)). Compare this expression with the one on the RHS in the question: on simple comparison, it can be observed that it is obviously bigger. However, since the arc of a circle is longer than a straight line for the same subtended angle, we can conclude that the LHS is bigger than the RHS.
To solve this you have to be intimately aware of standard theta value representations and cosines of pi.
I am not, so I didn't recognize the initial relationship between the 2 samples given.
I feel like this problem was kind of created through reverse engineering of sorts, haha.
Between this example then apply it to the interesting fact that the two towers on the Golden Gate Bridge are out of parallel by about 8" (if memory serves me correctly). This gives me the answer when a child asks that proverbial "Why do I need to learn this?" Thanks for sharing.
Calculator enjoying the content in a corner with puffs 😂😂😂
I had to use the brute force approach where i used the Pythagorean theorem in repeated application. First to compute the chord when it is 30 degrees , and then again for 15 degrees. Took me much longer than you guys but did not need any trig.
I considered one sector of a 24-gon and calculated from there. I had struggles with the square root though, as I kept using the different of angles law and not the double angle formula. So that was my mistake from there as I should have double angled pi/6 twice.
i havent watched the video but does it involve splitting a circle into 12 parts and drawing a triangle in one of the parts to approximate the area
I looked at it and said the left > right, but I could be wrong. Then broke the rules and did a quick estimate in my head and concluded the same. I relized they would be close to the same value, but not as close as it turned out to be. I haven't done trigonomitry by hand in over 30 years, so that didn't occur to me.
Hey, I like the simple way as solution. Suitable for A level
Wow I solved it by guess work but not sure if that's correct.. π can be written as 22/7 then the whole LHS side can be written as 2× (11/(7×6) ) so on LHS we have a quantity multiplied by two and on rha we have square root of 2 minus square root of something.. that's obviously going to be smaller ...
Well... the problem is that the error in the approximation of π as 22/7 is more than the given difference between the two quantities, and 22/7 > π
@@dlevi67 oki,,
At 2:30, why are the triangles labeled differently? The a, b, c, are different sides in each drawing. Doesn't that alter the calculations?
theta is gamma in this image
The only thing matters is that 'c' must be opposite to the angle you are taking the cos of
PI / 12 is .00074+ greater than the roots of roots of roots
gotta be one of the harder questions ever
My first thought was "Who cares?"
The I realised that someone must care to have asked the question.
I would assume that Pi/12 would still be a good sized portion, so I would go for that.
Clearly:
o
Vs
Root(1-o)
With a calculator, you would get facts: π/12 > 17/65 > 523/2000 > 59/226 > √(2 - √(2 + √3)) = 2 sin(π/24)
With reasoning, you would get proofs: x > 0 → x > sin(x) → x > 2 sin(x/2) ; x ∈ [0, π/2] → 2 sin(x/2) = √(2 - 2 √(1 - sin²(x)) ; 2 sin(π/24) = √(2 - 2 √(1 - sin²(π/12))) = √(2 - 2 √(1 - ((1/2) √(2 - 2 √(1 - sin²(π/6))))²)) = √(2 - √(2 + 2 √(1 - sin²(π/6)))) = √(2 - √(2 + 2 √(1 - 1/4))) = √(2 - √(2 + √3)) ; ∴ π/12 > √(2 - √(2 + √3)) □
But it takes experience to recognize that a certain expression in radicals might have a trigonometric representation. Thus "there is no royal road to geometry [meaning mathematics]".
Hahaha 😂 made it by evaluating it the hard way (without calculator, but with a bit of math intuition, and probably a lot of luck)
I cheated, I used a spreadsheet to find that pi()/12 looks larger by 0.000747 which is less than 0.001
...
*Cosine Rule
always the best
Is math discovered or invented.?
The # of sheep = the # of fingers is discovered.
This kind of stuff is invented.
the question is easy, really... a fraction of pi, is bigger than a negative number...
What negative number?
@@gavindeane3670 well, lets just take it numbers, without the square roots, since I don't know how to put them here. 2-(2+3) even if you do the squares, that's still 2-5... and since you go left to right, 2-5 is a negative number... square root of 2, minus the sum of the square root of 2 plus the square root of 3.... the part in the "nested" square is larger than the initial number... so... negative.
@@gwouru The part in the nested square root is not smaller than the initial number. There's no 2+3. It's 2+√3.
√3 is about 1.732, so 2+√3 is less than 4, so √(2+√3) is less than 2, so 2 - √(2+√3) is positive.
@@gavindeane3670 you missed something... and I'm going to copy your symbol to help... √2 - (√2+√3)
So... if we take it as is... with shortened numbers we get 1.414 - (1.414+1.732) so, that equals 1.414 - 3.146 = (-1.732)
Or simplify negative √3
@@gwouru Are you watching the same video as me???
There's no √2 +√3. It's √(2+√3). And at no point are we ever taking the square root of 2 on its own, so there's no 1.414 anywhere.
We have
√(2 - √(2 + √3))
√3 is about 1.732 so 2 + √3 is 3.732. So we have
√(2 - √(3.732))
√(3.732) is about 1.932, so we have
√(2 - 1.932)
2 - 1.932 is a positive number.
an electric train goes 100 miles per hour traveling a 200 miles distance west and the wind is 20 miles per hour east. how fast the train's steam smoke coming out of the train steam stack and why?
No steam. The train is electric.
@@stephenpeterkin correct. this was in a 1950's army recruitment iq test to measure common sense.
African or European tracks?
@@JohnWalker-e6y USA in preparations for the korean war in the 1950's.
3:58 just multiply by 0.5 both sides.
FO
there is no proof of why c
he did note that the shortest distance between two points is a straight line. c is a straight line and the theta arc isn't.
so that's not just an observation, just referring to a statement that doesn't really need proof (or if you want proof of that you can look it up).
@@teambellavsteamalice I am just kidding，I think it can't be proved, it's just an axiom.
@@user_uif_ghg_wer_das of course you can prove the shortest distance between two points on a flat surface is a straight line, calculus of variations is your friend there.
@@user_uif_ghg_wer_das
Isn't there a formula for the circumference of polygons?
Like an n-sided one has n times a formula like in this video. Then you could have a limit where n goes to infinity of this length steadily increasing and approaching 2π.
@@user_uif_ghg_wer_das It can be proved in multiple ways. I think the most rigorous one uses differential geometry.
The right one is larger.
I don't really know, but the confidence sells it.
If it turns out to be right, I just sit back with the slightest of smiles.
If it turns out I'm wrong, I just say, "Ohhh, you think the right is on THAT side.", and then they start questioning everything all over again.
It's the left one
I used calculator to check
aww, you had a 50% chance!
my guess also, after I wasted some time figuring out (4+2√3) = (1+√3)^2 so the inner root can be simplified a little bit. and then I got stuck.
so, you saved yourself some time, very efficient!
@@Maanjiro_g right
only a buncha math geeks can be intersted in this one.
And here we are. Welcome!
If difference is 0.00099(9) does it less than 0.001?
No.