Unanswered Question: TanX - SinX = ?

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Danang

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Unanswered Question: TanX - SinX = ?
« on: September 24, 2018, 03:43:34 AM »
Let me copypaste:

"When sin points out vertically to the arch segment's pointer edge, and tan is what the pointer faces at the coordinate X line,
according to school's math, how do you meassure the difference between tan and sin, or the tanX°-sinX°?
Exp.: the difference for 30° = tan30° - sin30°= 0.57735-0.5= 0.07735
Is there an equation for such length?

Let me check whether it is similar to my finding about it. (I'll post after the reply)"
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Re: Unanswered Question: TanX - SinX = ?
« Reply #1 on: September 24, 2018, 09:13:17 AM »
Let me copypaste:

"When sin points out vertically to the arch segment's pointer edge, and tan is what the pointer faces at the coordinate X line,
according to school's math, how do you meassure the difference between tan and sin, or the tanX°-sinX°?
Exp.: the difference for 30° = tan30° - sin30°= 0.57735-0.5= 0.07735
Is there an equation for such length?

Let me check whether it is similar to my finding about it. (I'll post after the reply)"

Yes. The equation that will give the difference, D, between the tangent and sine of an angle, x, is:

D(x) = tan(x) - sin(x) [Eq. 1]

If, for some reason you're averse to using sin(x) and remember that tan(x) = sin(x)/cos(x), you can restate that as:

D(x) = tan(x) - sin(x)
 = sin(x)/cos(x) - sin(x)
 = sin(x)/cos(x) - cos(x)sin(x)/cos(x)
 = sin(x)/cos(x) (1 - cos(x))
D = tan(x) (1 - cos(x))  [Eq. 2]
D = tan(x) - tan(x)cos(x)  [Eq. 3]

Alternatively, since sin2(x) = 1 - cos2(x), you could rewrite it as:

D =  tan(x) - (1 - cos2(x))1/2  [Eq. 4]

I'm not seeing a big win there, though. Equation 1 is simpler than equations 2, 3, and 4.
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sandokhan

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Re: Unanswered Question: TanX - SinX = ?
« Reply #2 on: September 24, 2018, 09:27:44 AM »
Equations 2, 3 and 4...


Here is how it's done.

tan x - sin x = sin x (1/cosx - 1) = sinx (1/{1/2 x (({ [( (2 - x2/2N)2 - 2)2...]2 - 2}2 - 2))} - 1)

Re: Unanswered Question: TanX - SinX = ?
« Reply #3 on: September 24, 2018, 11:51:55 AM »
tan x - sin x = sin x (1/cosx - 1) = sinx (1/{1/2 x (({ [( (2 - x2/2N)2 - 2)2...]2 - 2}2 - 2))} - 1)

That's even more complicated. Congratulations!
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sandokhan

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Re: Unanswered Question: TanX - SinX = ?
« Reply #4 on: September 24, 2018, 01:00:44 PM »
Not at all.

For angles less than π, N = 6, 3 + 1 evaluations.

That is, my formula for the cosine is a GLOBAL formula, not a local formula like the Taylor expansions.

Then, since sinx = ±(1 - cos2x)1/2, the entire approximation can easily be evaluated.

One might also use the following identity:

cos x = sin x (1/x + 2x/(x2 - π2) + ...)

Re: Unanswered Question: TanX - SinX = ?
« Reply #5 on: September 24, 2018, 02:40:52 PM »
Not at all.

For angles less than π, N = 6, 3 + 1 evaluations.

Eq. 1 requires two (2) evaluations.

Quote
One might also use the following identity:

cos x = sin x (1/x + 2x/(x2 - π2) + ...)

Perhaps, but I don't see why one would want to.

The OP has some replies. Let's see if any are useful to him.
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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #6 on: September 25, 2018, 01:26:37 AM »
Thank you Alpha2Omega & Sandokhan.

Maths is actually interesting as long as it leads one to the correct calculation.

Moreover if it can be explained visually with a diagram or graphic.

Perhaps an explanation can boost imagination i.g. the moving tan be substracted by some length to result the difference between tan and sin as above.
And for bigger scoup, when that difference is added with sin, it will result the value of tan. So we know within particular angle, what is contributed by sin and what is contributed by the difference in forming tan's length.

Mine, while sin is moving, there is additional value accompanying  i.e. the difference, so as to form a length of tan.

TanX = SinX + Difference SinX & TanX
= SinX + SinX*Diagonal.
= SinX + SinX * √[(tanX-sinX)²+(1-cosX)²]

In this formulae, while sin is moving, you can imagine how the diagonal grows from zero to maximum 0.4142 within the corner of 0.2929 * 0.2929.
The maximum value of the diagonal i.e. 0.4142 has the limit due to the muliplier of sin45 i.e. 0.7071. So the maximum difference within 45°=0.7071*0.4142=0.2929.
Tan45=0.7071+0.2929=1.
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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #7 on: September 25, 2018, 01:31:25 AM »
Now I wanna know from you about Pi=3.14159.

Are you sure that's the correct value? If so, please explain.
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Re: Unanswered Question: TanX - SinX = ?
« Reply #8 on: September 25, 2018, 01:50:05 AM »
Now I wanna know from you about Pi=3.14159.

Are you sure that's the correct value? If so, please explain.
So you'll just ignore the direct questions I put to you in the other thread.  Just goes to show how little confidence you have in your silly phew.
Since it costs 1.82¢ to produce a penny, putting in your 2¢ if really worth 3.64¢.

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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #9 on: September 25, 2018, 03:22:21 AM »
You wanna answer, Mike?
Go for it.  8)
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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #10 on: September 25, 2018, 03:23:17 AM »
I always be confident with Phew.  8)
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rvlvr

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Re: Unanswered Question: TanX - SinX = ?
« Reply #11 on: September 25, 2018, 04:12:24 AM »
Phew is best!

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rabinoz

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Re: Unanswered Question: TanX - SinX = ?
« Reply #12 on: September 25, 2018, 04:23:41 AM »
Now I wanna know from you about Pi=3.14159.

Are you sure that's the correct value? If so, please explain.
That's an approximation to π = circumference/diameter of a circle.

You did, I suppose, read Flat Earth Debate / Re: Tangent & C Relation: 3.14159 vs Phew's 3.17157 « on: Today at 08:00:05 AM ».
In that:
Way back around 250 BC old Archimedes did better than you:
Quote from: Wikipedia
Pi, Antiquity
Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (that is 3.1408 < π < 3.1429).
Even that is far closer to the correct value of than π than your Phew =  3.17157.

Archimedes reasoned that the π can be estimated by computing the perimeters of circumscribed and inscribed polygons.

He started with 6-sides and found an "easy" way to calculate the new circumferences each time the number was doubled.

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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #13 on: September 25, 2018, 04:53:23 AM »
Whenever there is anyone that mentions the word "approximation" in figuring out C/D, I've got only two words to say:

"Go Home"   ;D  ;D  ;D
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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #14 on: September 25, 2018, 05:00:17 AM »
Series: the more you add it, the more uncounted spaces showing up.

When will you reach happiness if you're always stuck on series??  :o   8)
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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #15 on: September 25, 2018, 05:08:07 AM »
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rabinoz

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Re: Unanswered Question: TanX - SinX = ?
« Reply #16 on: September 25, 2018, 05:19:44 AM »
Whenever there is anyone that mentions the word "approximation" in figuring out C/D, I've got only two words to say:
Any measurement you make is an approximation to the exact value.
Any value you quote for π is an approximation, even if I say that:
π =3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
     8214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196
     4428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273
     7245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094
     3305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912
     9833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132
     0005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235
     4201995611212902196086403441815981362977477130996051870721134999999837297804995105973173281609631859
     5024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303
     5982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989

Even though that is π to 1000 places, it's still an approximation.

If you don't like that, tough!

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rvlvr

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Re: Unanswered Question: TanX - SinX = ?
« Reply #17 on: September 25, 2018, 05:20:33 AM »
Phew is best!

Please tell me why.
I can't put it any better than this!

Seems to suffer from rounding errors for extremely small radii that get worse as radius increases.

Phew.. ^_^

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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #18 on: September 25, 2018, 05:23:54 PM »
I proposed different Phews till the last and convincing Phew: 3.17157 with clear calculations and supported by numerous diagram mindblowing hints.

Stick to 3.17157 and skip the rest.
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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #19 on: September 25, 2018, 05:29:20 PM »
Here's another hints:
Phew/2 = r + (45°diagonal)^2
= 1 + 0.765366^2
= 1.58578
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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #20 on: September 25, 2018, 05:35:12 PM »
"Any measurement you make is an approximation to the exact value."

>> the problem is: you regard the series edge as a basis to form pi.
This apporoximation is irrelevant. The deviation from the true C is much visible than something in between sin & tan. That's why Jack feels horrible when performing pi experiment. :o
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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #21 on: September 25, 2018, 05:46:19 PM »
Here's another hints:
Phew/2 = r + (45°diagonal)^2
= 1 + 0.765366^2
= 1.58578

Of course for the correct dimension, it should be :

Phew/2 = r^2 + (45°diagonal)^2
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Stash

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Re: Unanswered Question: TanX - SinX = ?
« Reply #22 on: September 25, 2018, 06:37:30 PM »
Mine, while sin is moving, there is additional value accompanying  i.e. the difference, so as to form a length of tan.

I'm pretty sure this is the best part, the made up part (bolded).

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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #23 on: September 25, 2018, 07:20:17 PM »
What did you mean, Stash?
Everything goes well with my formulae, no error in such calculation. Only tried to provide the better n nicer look for easier comprehending.
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Stash

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Re: Unanswered Question: TanX - SinX = ?
« Reply #24 on: September 25, 2018, 10:45:26 PM »
What did you mean, Stash?
Everything goes well with my formulae, no error in such calculation. Only tried to provide the better n nicer look for easier comprehending.

Yes, where exactly did this come from (bolded):

Here's another hints:
Phew/2 = r + (45°diagonal)^2
= 1 + 0.765366^2
= 1.58578


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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #25 on: September 25, 2018, 11:06:59 PM »


In a square of 0.7071×0.2929, a 45° arch stays. The length of the diagonal within this square equals 0.765366.
The scale: r=1.
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Stash

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Re: Unanswered Question: TanX - SinX = ?
« Reply #26 on: September 25, 2018, 11:32:07 PM »


In a square of 0.7071×0.2929, a 45° arch stays. The length of the diagonal within this square equals 0.765366.
The scale: r=1.

And how did you decide on this measurement of a square: 0.7071×0.2929?

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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #27 on: September 25, 2018, 11:51:21 PM »
"And how did you decide on this measurement of a square: 0.7071×0.2929?"

>> I have no idea yet. Research is on the go. It's just a numbers game within a 1×1 square with arch and intersections.
I regard it as hints for the arch length, including phew.
There are numbers combinations by 0.2929, 0.7071, 0.76536, 0.4142, 1, 1.4142 etc that form an arch length with various special angles: 45° & 90° which in turn can determine the exact phew value.

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Danang

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Re: Unanswered Question: TanX - SinX = ?
« Reply #28 on: September 25, 2018, 11:58:26 PM »
0.7071^2+0.2929=0.7929
0.7071+0.2929^2=0.7929
1 + [ (1+1) - 1.4142 ] = 1.5858
1.4142 + 0.4142^2 = 1.5858
etc. etc.

There are so many hints to confirm phew=3.17157.

You too can figure out the rest.
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Stash

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Re: Unanswered Question: TanX - SinX = ?
« Reply #29 on: September 26, 2018, 12:09:45 AM »
0.7071^2+0.2929=0.7929
0.7071+0.2929^2=0.7929
1 + [ (1+1) - 1.4142 ] = 1.5858
1.4142 + 0.4142^2 = 1.5858
etc. etc.

There are so many hints to confirm phew=3.17157.

You too can figure out the rest.

How did you derive 0.7071^2+0.2929 from a "1×1 square"? What was your starting value of '1x1'?