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391:"Proposition 12 In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the EITHER OF THE squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that RECTANGLE on WHOSE LENGTH the SIDE SUBTENDING the perpendicular falls, and WHOSE WIDTH IS EQUAL TO the straight line SEGMENT cut off outside by the perpendicular towards the obtuse angle."
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unclear if by 'that' he means namely, 'the rectangle' or namely, 'the side about the obtuse angle' which he is describing. It seems to me either Euclid was not expressing himself clearly here or the translation by Heath is at fault but the end of the Euclid quote does not seem to describe a rectangle very well. It might have been clearer if Euclid had stated something closer to the following (CAPS mark my paraphrasing):
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1521:.In Fig 1 triangle ABC is drawn with B an acute angle. With C as centre and BC (=a) as radius semi-circle DBF is constructed with DF a diameter and D a point on AC. The circle meets side AB at E and EC is joined forming isosceles triangle ECB with EB = 2acos(B).Power of Points Theorem (intersecting secants theorem) is applied to point A outside the circle:
3556:
3325:{\displaystyle {\begin{aligned}c^{2}&=(a+b)^{2}-4ab\cos ^{2}{\tfrac {\theta }{2}}\\&=4m_{a}^{2}-4m_{g}^{2}\cos ^{2}{\tfrac {\theta }{2}}\\&=4(m_{a}+m_{g}\cos {\tfrac {\theta }{2}})(m_{a}-m_{g}\cos {\tfrac {\theta }{2}})\\&=(2m_{a}+2m_{g}\cos {\tfrac {\theta }{2}})(2m_{a}-2m_{g}\cos {\tfrac {\theta }{2}})\\\end{aligned}}}
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If you define the dot product (a,b)*(c,d) as ac+bd (so it is linear) and then prove (a,b)*(c,d)=|(a,b)|*|(c,d)|*cos(α) then there is nothing circular. The definition used is a matter of taste, I guess some others are possible. I like the argument with dot product because it gives a different light on
1454:
The last demonstrations using the power of a point can be simplified, all of them can be worked out with just one application of the power point theorem, and the the first two cases can be made one by considering b<c(always can be done by flipping the sides). Someone willing to make new pictures?
1430:
I have rewritten the article and expanded it considerably. I have taken a lot of material from the French article, according to the above suggestion. I hope you like it!!! Please improve further... (and sorry that in the history of this page the big change was signed "Euklid". That was me and and
387:
The text quoted by Euclid concerning the geometric interpretation of the Law of
Cosines is helpful but unclear in the sense that Euclid (or possibly Heath) did not name the thing Euclid meant to "namely" identify. In the quote Euclid states 'namely "that" on which the perpendicular falls', but it is
5560:
Could there be a valid proof using the definition of a dot product. With triangle having side C as the vector A-B, where A,B are the vectors of the triangle legs: then length |C|^2 could be written as |A|^2+|B|^2-2A(dot)B. A(dot)B is defined as |A||B|cos(theta) giving the law of cosines formula. If
4263:{\displaystyle {\begin{aligned}c^{2}&{}=(b\cos \theta -a)^{2}+(b\sin \theta )^{2}\\c^{2}&{}=b^{2}\cos ^{2}\theta -2ab\cos \theta +a^{2}+b^{2}\sin ^{2}\theta \\c^{2}&{}=a^{2}+b^{2}(\sin ^{2}\theta +\cos ^{2}\theta )-2ab\cos \theta \\c^{2}&{}=a^{2}+b^{2}-2ab\cos \theta .\end{aligned}}}
5661:
The angle ABC in Fig 2 clearly looks acute on my monitor. Just to be clear an acute angle is smaller than 90 degrees and an obtuse angle is greater than 90 degrees but less than 180 degrees. It makes the whole section unreadable, and brings into question the correctness of articles on the site.
1376:
As noted above, the vector-based proof is circular: the proof that the dot product of two vectors has a geometric interpretation in
Euclidian space is itself based on the law of cosines. So using the geometric interpretation of a dot product to prove the law of cosines is a bit problematic...
418:
In "Using the distance formula" the article states as a fact that "We can place this triangle on the coordinate system by plotting..." and then proceeds to use some trigonometry in the coordinates, without any explanation for how those coordinates are derived / where they came from.
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I moved the following here to the talk page. This proof combines trigonometry and the
Pythagorean theorem and quite a bit of algebra of real numbers. It is not nearly as elementary as some of the other proofs presented. I don't see what is gained by including it in the
4283:
Why banish this one? It sets up one equation and grinds it out. The trig proof that is left requires looking at the triangle in three different ways to come up with three different equations to mash together. It, too, seems to involve "quite a bit of algebra of real
2593:
2127:{\displaystyle {\begin{array}{lcl}\quad \quad AD\times AF=AB\times AE\\\Rightarrow (b-a)(b+a)=c(c+2a\cos(180-{\hat {B}}))\\\Rightarrow \quad b^{2}-a^{2}=c^{2}-2ac.\cos {\hat {B}}\\\Rightarrow \quad \quad b^{2}=a^{2}+c^{2}-2ac\cos {\hat {B}}\end{array}}}
2502:{\displaystyle {\begin{array}{lcl}\quad \quad BD\times BE=BC\times BF\\\Rightarrow (b-c)(b+c)=a(2b\cos {\hat {C}}-a)\\\Rightarrow \quad b^{2}-c^{2}=2ab.\cos {\hat {C}}-a^{2}\\\Rightarrow \quad \quad c^{2}=b^{2}+a^{2}-2ab\cos {\hat {C}}\end{array}}}
1813:{\displaystyle {\begin{array}{lcl}\quad \quad AD\times AF=AB\times AE\\\Rightarrow (b-a)(b+a)=c(c-2a\cos {\hat {B}})\\\Rightarrow \quad b^{2}-a^{2}=c^{2}-2ac.\cos {\hat {B}}\\\Rightarrow \quad \quad b^{2}=a^{2}+c^{2}-2ac\cos {\hat {B}}\end{array}}}
2547:
1) The angle at vertex C is acute, and the angle at vertex B is obtuse. a - b*cos(theta) becomes b*cos(theta) - a. (where as in the original theta is the angle at vertex C). Since the term is squared the result is the same since x**2 = (-x)**2
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changed the article. I suggest that someone reverts the change, unless there is consensus to do otherwise. I have already reverted a couple of times, but don't want to engage in an edit war, so I'm taking the article off my watchlist.
4599:
All triangles have three Sines, three
Cosines, and three Tangents. The same goes for their inverses too. The article's triangular drawing displays three angles, therefore three Cosine identies exists, as I have shown above ... Mike
725:
The dot product proof should be removed. It's circular. Law of cosines comes first historically and all of vector calculus is assumed to depend on it, not vice versa. Might as well use vectors to prove the pythagorean theorem.
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of Book 1 Copernicus describes two techniques for determining angles given all three sides of a triangle. These techniques correspond respectively to the
Pythagorean and Power of Points derivations of the Law of Cosines.
691:
It seems that your answer would technically be correct, Michael, but it's a bit unclear because the variables tend to be such that A and B are the legs and C is the hypotenuse. It's not a correctness issue, it's clarity.
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If noone objects, I'm going to put the vector-based proof first and move the other one down since the former is more simple and universal as opposed to the latter.. or someone else could do it.. or whatever... -
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The diagram and symbols used for angles on the "Law of cosines" page are not the same as those on the "Law of sines" page. Would it be reasonable to change this page to match those on the "Law of sines" page?
3551:{\displaystyle {\begin{aligned}({\tfrac {c}{2}})^{2}&=m_{a}^{2}-m_{g}^{2}\cos ^{2}{\tfrac {\theta }{2}}\\&=(m_{a}+m_{g}\cos {\tfrac {\theta }{2}})(m_{a}-m_{g}\cos {\tfrac {\theta }{2}})\\\end{aligned}}}
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Triangle ABC is drawn with side AB=c,BC=a and CA=b. With A as centre construct circle DCE with radius b and diameter DE passing through B. CB produced meets the circle at F. Since triangle CAF is isosceles:
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This proof along with that using
Pythagoras theorem are of considerable historical significance since both have their origins in Euclid's Elements and both are referred to in the ground breaking work of
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Great work!! The French article was way better by any standards, I fell bad I can't understand it. I only see the general flow of ideas. Now it is all clear I will read in finer detail, Thanks!!!
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I changed the line "CH = a cos(π – γ) = −a cos(γ)" in the history subsection to "CH = (CB) cos(π – γ) = −(CB) cos(γ)". In the equations used in this section and the diagram there is no "a" side.
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5426:- If c=b, there is one solution. It is an isosceles triangle with b=c, B=C. The triangle itself is twice the size as the right triangle formed by c=b*sin(C) & B = 90. B and C are acute.
5422:
In the section where you use the quadratic equation to make a form of the law of cosines that solves angle-side-side triangles, there is an error on the bounds checking. You say that if c: -->
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In addition to the two cases dealt with above, we also need to consider the situation shown in the diagram where Power of Point theorem is applied about point B inside the construction circle.
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2809:{\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}-2ab\cos \theta \\&=(a+b)^{2}-4ab\cos ^{2}{\tfrac {\theta }{2}}\\&=(a-b)^{2}+4ab\sin ^{2}{\tfrac {\theta }{2}}\\\end{aligned}}}
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231:. I'm no mathematics scholar, but I'm quite sure that isn't the theorem it speaks of. I could be dead wrong, I'm not the most educated fellow, but I think it's worth some speculation.
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I will ignore cases when any of the angles are right angles, as then the distance formula simply leads to the
Pathagorian Theorem, which is a special case of the law of cosines.
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Will it make sense to add a demonstration with
Chasles relation and scalar product? Even if historically it was after, it is a simple and powerful tool to demonstrate it.
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I never saw the story of the dot product but I would guess it came from the scalar product of physics (as in the definition of work done in a particle) is that right?
902:
if you know the length of the three sides. This is how you do it:            
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article, a dot b = a b cos theta is proved using the Law of
Cosines. And the Law of Cosines is proved using vector dot products. Shouldn't somebody fix this? --
35:
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I guess we could also define (a,b)*(c,d) as |(a,b)|*|(c,d)|*cos(α) prove the linearity by geometrical arguments and use it to prove the cosine law. right?
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441:
Okay, in the article, it states the since cos 90 is 0 the Law of
Cosines is reduced to the Pythagorean Theorem. I don't think that's the case cause if
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Of course then you have to show why A dot B = |A| |B| cos (theta) is a reasonable definition, since usually dot products are defined by coordinates.
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Is the usage of "we" throughout this article proper? Shouldn't "we can easily prove" be "can be proved" (wlong with some sentence rearrangement).
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P.S. I only place this talk section above the others to more closely align it with the portion of the main article to which it pertains.
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11-03-09: Apparently NorweigianBlue and HambergerRadio don't know their Trigonometry. There are three Cosine formulas, NOT one formula.
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I have hopefully solved this problem, by stating the law of cosines is equivalent to the dot product formula from theory of vectors. --
394:
I am not suggesting that the quote be changed but only that a clarification be given by somebody more in the know on this than myself.
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Right now it says, "only one positive solution if c = b sin γ". It should say "only one positive solution if c = b sin γ OR if c: -->
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An advantage of this proof is that it does not require the consideration of different cases for when the triangle is acute vs. obtuse.
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If you Google "Law of Cosines", you will find the above three formulas, which corresponts to the article's triangle figure.
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I really like this proof, but it seems to be there are multiple cases. All of which work out to the same correct result.
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Yes, and that's the Pythagorean theorem (in the case where the right angles is the angle between the sides of lengths
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I don't see how that puts my argument down, however, I do not think that it is reasonable for me to argue about this.
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The proof given works when angle at vertex C is acute and angle at vertex B is acute. Below are the other cases:
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I removed the following section because this is said in the first paragraph and follows from elementary algebra:
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The alternate forms were found after reading about rounding errors in spherical triangles, as mentioned in the
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268:. Standardization (i.e.: Mathematics)— given a right triangle, the square of the length of the hypothenuse
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Perhaps i'm mistaken but angle ABC in Fig 2 appears acute rather than obtuse to me. Proof reading anyone?
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4779:, so stating the formula just once covers all three sides of the triangle (and also all other triangles).
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No, it's definitely obtuse (unless there's some issue with it being displayed in a weird aspect ratio). –
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Now apply the Power of a Point Theorem (intersecting chords theorem) to point B inside the circle:
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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754:"a dot b = a b cos theta" is the definition of a vector dot product. No proof is not required.
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Right, ABC is an angle not a triangle. 'Triangle ABC' is confusing if not complete nonsense.
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Just a comment about the readability- I can't actually see clearly the labels on Figure 2.
272:, opposite to the right angle, equals the sum of the squares of the lengths of the two legs
2918:{\displaystyle {\begin{aligned}m_{a}&=(a+b)/2\\m_{g}&={\sqrt {ab}}\\\end{aligned}}}
173:
The bit in the Ptolemy's theorem section references the Pythagorean theorem, citing it as
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I know we shouldn't use these kinds of names on Knowledge, so I've changed my login. --
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equal to a + b*(-cos(theta)) or a - b*cos(theta). Again leading to the same result.
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b, there is also one solution. It is a scalene triangle with B and C being acute.
3751:{\displaystyle A=(b\cos \theta ,\ b\sin \theta ),\ B=(a,0),\ {\text{and}}\ C=(0,0).}
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If you need any diagrams drawn, altered, fixed etc. Please post a request at the
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The Law of Cosines can also be used to find the measure of the three angles in a
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Replica of diagrams in Book 1, Page 21 of "De Revolutionibus Orbium Coelestium"
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1468:. Please read instructions at the top of the page before posting. Thank you!
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4626:. The law is stated there only once, not three times as is it is here after
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the other two sides. Thus it can be applied first with one side labeled
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All three are the same formula. Therefore there is just one Cosine Law.
3858:{\displaystyle c={\sqrt {(b\cos \theta -a)^{2}+(b\sin \theta -0)^{2}}}.}
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5506:. Also, please place new comments at the end and sign your posts. See
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It will be no different in the case of Fig 2 where angle B is obtuse:
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other editors think this is valid, I could add a formatted proof.
5244:{\displaystyle {\frac {\sin C}{\frac {1}{6.25/2}}}={\sqrt {5}}.25}
3632:. We can place this triangle on the coordinate system by plotting
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The arithmetic and geometric mean lengths of the known sides are:
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1-0.68=0.32 change between the law of sin and the law of cosine
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the cosine law and I think it deserves a place on this article.
5129:{\displaystyle {\frac {\sin A}{\frac {1}{2.5}}}={\sqrt {5}}.25}
5180:{\displaystyle {\frac {\sin B}{\frac {1}{2.5}}}{\sqrt {5}}.25}
15:
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Intended meaning of Euclid's Proposition should be made clear
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is the measurement of the angle opposite the side of length
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2) The angle at vertex C is obtuse. in this case the side:
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Adjusted line "CH = a cos(π – γ) = −a cos(γ)" in history
4622:
In the French Knowledge, the Law (singular) of cosines,
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a - b*cos(theta) becomes a + b*cos(180-theta) which is
670:. I put something about this near the top of the page.
246:
a²+c²=b² is the Pythagorean theorem. What do you mean?
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means: the triangle defined by the vertices A, B, C.
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ABC is obtuse, because angle ACB is an obtuse angle. –
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Another small addition on the angle-side-side solution
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Proof of Law of Cosines using Power of a Point Theorem
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Demonstration by Chasles relation and scalar product
4735:{\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos(\alpha )\,}
4559:{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos(\gamma )\,}
4393:{\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos(\alpha )\,}
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The last time I checked, the Pythagorean theorem is
112:, a collaborative effort to improve the coverage of
4476:{\displaystyle b^{2}=a^{2}+c^{2}-2ac\cos(\beta )\,}
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5600:Dot product#Scalar projection and first properties
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1151:{\displaystyle \cos C=(-a^{2}-b^{2}+c^{2})/(-2ab)}
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437:Pythagorean Theorem also Proved by Law of Cosines?
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5418:Correction for solving angle-side-side triangles
3576:Proof using distance formula moved to talk page.
1243:{\displaystyle \cos C=(a^{2}+b^{2}-c^{2})/(2ab)}
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5812:Knowledge level-5 vital articles in Mathematics
1053:{\displaystyle -2ab\cos C=-a^{2}-b^{2}+c^{2}}
8:
4938:{\displaystyle \arccos {\frac {4.25}{6.25}}}
1493:: "De Revolutionibus Orbium Coelestium". On
4293:I don't see what is gained by moving it. -
972:{\displaystyle a^{2}+b^{2}-2ab\cos C=c^{2}}
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4840:relation between the law of sin and cosine
2199:{\displaystyle CF=2b\cos {\hat {C}}\quad }
414:Proof "Using the distance formula" unclear
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1351:Now you can find the measure of angle C!
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511:{\displaystyle a^{2}=b^{2}+c^{2}-2bccosA}
478:
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210:
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4904:{\displaystyle \arccos {\frac {1}{2.5}}}
4870:{\displaystyle \arccos {\frac {1}{2.5}}}
224:{\displaystyle a^{2}+c^{2}=b^{2}.\quad }
5802:Knowledge vital articles in Mathematics
5670:2605:A601:A706:8700:F91E:CE18:C1EC:7244
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330:{\displaystyle c^{2}=a^{2}+b^{2}\quad }
60:
19:
5423:=b, there are no solutions. However:
4311:Law of Cosines is plural, NOT singular
2560:Does this seem reasonable to anyone?
5817:C-Class vital articles in Mathematics
3871:Now, we just work with that equation:
7:
1513:Fig 1 and Fig 2 are replicas of the
106:This article is within the scope of
5556:Proof by application of dot product
4813:Make consistent with "Law of sines"
4775:, then with the third side labeled
1519:De Revolutionibus Orbium Coelestium
49:It is of interest to the following
5827:High-priority mathematics articles
5737:
5683:Oh, I think I see the confusion.
5376:{\displaystyle 1.5\times 3.5=5.25}
5070:{\displaystyle 1.5\times 3.5=5.25}
4771:, then with a second side labeled
14:
5584:Absolutely. It should be added --
663:{\displaystyle a^{2}+b^{2}=c^{2}}
564:{\displaystyle a^{2}=b^{2}+c^{2}}
126:Knowledge:WikiProject Mathematics
5797:Knowledge level-5 vital articles
5343:{\displaystyle {\frac {1}{2.5}}}
5037:{\displaystyle {\frac {1}{2.5}}}
3764:By the distance formula, we have
789:Wow that's pretty darn smart. --
129:Template:WikiProject Mathematics
93:
83:
62:
29:
20:
146:This article has been rated as
5807:C-Class level-5 vital articles
5705:23:28, 27 September 2020 (UTC)
5678:23:19, 27 September 2020 (UTC)
5657:16:55, 22 September 2020 (UTC)
5637:16:24, 22 September 2020 (UTC)
5315:{\displaystyle 4.25,5.25,6.25}
5009:{\displaystyle 4.25,5.25,6.25}
4727:
4721:
4641:All three say the same thing.
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720:07:48, 9 September 2005 (UTC)
325:
256:13:41, 24 November 2019 (UTC)
219:
120:and see a list of open tasks.
5822:C-Class mathematics articles
5594:18:37, 10 October 2020 (UTC)
5524:12:44, 26 January 2020 (UTC)
5497:06:06, 26 January 2020 (UTC)
5470:01:51, 26 January 2020 (UTC)
5446:01:35, 26 January 2020 (UTC)
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4616:00:54, 4 November 2009 (UTC)
4594:23:27, 3 November 2009 (UTC)
351:21:40, 3 December 2019 (UTC)
5725:23:35, 2 October 2020 (UTC)
5282:{\displaystyle 1.5,2.5,3.5}
4976:{\displaystyle 1.5,2.5,3.5}
4759:the angle opposite it, and
409:17:48, 27 August 2012 (UTC)
5843:
5752:{\displaystyle \Delta ABC}
5730:It's perfectly clear what
5579:21:40, 6 August 2020 (UTC)
1460:08:02, 16 April 2007 (UTC)
1445:23:42, 12 April 2007 (UTC)
1436:10:45, 30 March 2006 (UTC)
1421:04:00, 30 March 2006 (UTC)
1395:10:45, 30 March 2006 (UTC)
874:23:20, 12 April 2007 (UTC)
851:23:20, 12 April 2007 (UTC)
828:23:20, 12 April 2007 (UTC)
744:10:45, 30 March 2006 (UTC)
432:20:52, 19 April 2011 (UTC)
373:21:45, 22 April 2020 (UTC)
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5611:17:44, 15 June 2024 (UTC)
5550:17:49, 15 July 2020 (UTC)
2580:01:30, 12 July 2008 (UTC)
1361:21:00, 25 Feb 2004 (UTC)
1341:{\displaystyle C=\arccos}
609:23:20, 1 March 2007 (UTC)
145:
78:
57:
4755:of the three sides, and
1483:Power of a Point Theorem
1481:Alternative proof using
1371:00:33, 22 May 2005 (UTC)
806:04:04, 21 May 2006 (UTC)
780:17:16, 20 May 2016 (UTC)
764:17:16, 20 May 2016 (UTC)
152:project's priority scale
5617:Requires proof reading?
3621:{\displaystyle \theta }
702:13:10, 6 May 2015 (UTC)
680:13:08, 6 May 2015 (UTC)
241:13:05, 6 May 2015 (UTC)
169:Pythagorean Theorem bit
109:WikiProject Mathematics
5792:C-Class vital articles
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101:Mathematics portal
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2012:
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1188:
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1021:
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1007:
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989:
979:
966:
962:
958:
955:
952:
949:
946:
943:
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937:
932:
928:
924:
919:
915:
895:
892:
887:
886:Section moving
884:
883:
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881:
880:
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860:
859:
858:
857:
856:
855:
854:
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837:
836:
835:
834:
833:
832:
831:
830:
813:
812:
811:
810:
809:
808:
769:
768:
767:
766:
749:
748:
747:
746:
734:
733:
708:
705:
689:
688:
687:
686:
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684:
657:
653:
649:
644:
640:
636:
631:
627:
612:
611:
558:
554:
550:
545:
541:
537:
532:
528:
507:
504:
501:
498:
495:
492:
489:
486:
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477:
473:
468:
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455:
451:
438:
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412:
384:
381:
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375:
356:
355:
354:
353:
322:
318:
314:
309:
305:
301:
296:
292:
281:
259:
258:
218:
213:
209:
205:
200:
196:
192:
187:
183:
170:
167:
164:
163:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
5839:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5808:
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5789:
5787:
5770:
5766:
5762:
5746:
5743:
5740:
5729:
5728:
5726:
5722:
5718:
5717:198.24.255.98
5714:
5708:
5707:
5706:
5702:
5698:
5694:
5693:Deacon Vorbis
5688:
5684:
5682:
5681:
5679:
5675:
5671:
5667:
5660:
5659:
5658:
5654:
5650:
5646:
5645:Deacon Vorbis
5642:
5641:
5640:
5638:
5634:
5630:
5629:198.24.255.98
5626:
5616:
5612:
5608:
5604:
5601:
5597:
5595:
5591:
5587:
5586:83.56.100.228
5583:
5582:
5581:
5580:
5577:
5576:
5555:
5553:
5551:
5547:
5543:
5539:
5529:
5525:
5521:
5517:
5513:
5512:Deacon Vorbis
5509:
5505:
5502:
5501:
5500:
5498:
5494:
5490:
5486:
5475:
5471:
5467:
5463:
5459:
5458:Deacon Vorbis
5455:
5451:
5450:
5449:
5447:
5443:
5439:
5435:
5427:
5424:
5417:
5415:
5414:
5410:
5406:
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5395:
5391:
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5364:
5361:
5358:
5351:
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5332:
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5309:
5306:
5303:
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5297:
5290:
5276:
5273:
5270:
5267:
5264:
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5256:
5255:
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5221:
5217:
5213:
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5198:
5188:
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5158:
5153:
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5096:
5093:
5083:
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5081:
5064:
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5058:
5055:
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4970:
4967:
4964:
4961:
4958:
4951:
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4949:
4930:
4927:
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4919:
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4888:
4885:
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4859:
4854:
4851:
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4839:
4837:
4835:
4831:
4827:
4823:
4812:
4810:
4809:
4805:
4801:
4793:
4791:
4790:
4786:
4782:
4781:Michael Hardy
4778:
4774:
4770:
4766:
4762:
4758:
4754:
4750:
4743:
4724:
4718:
4715:
4712:
4709:
4706:
4703:
4698:
4694:
4690:
4685:
4681:
4677:
4672:
4668:
4660:
4658:
4657:
4656:
4655:The identity
4653:
4652:
4648:
4644:
4643:Michael Hardy
4637:
4634:
4633:NorwegianBlue
4629:
4625:
4621:
4620:
4619:
4617:
4613:
4609:
4605:
4595:
4591:
4587:
4586:
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4542:
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4514:
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4505:
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4431:
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4339:
4335:
4330:
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4318:
4317:
4316:
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4300:
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4292:
4291:
4290:
4289:
4282:
4281:
4280:
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4274:
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4250:
4247:
4244:
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3691:
3686:
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3603:
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3297:
3292:
3288:
3284:
3281:
3276:
3272:
3268:
3256:
3253:
3247:
3244:
3239:
3235:
3231:
3228:
3223:
3219:
3215:
3209:
3207:
3192:
3189:
3183:
3180:
3175:
3171:
3167:
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3101:
3099:
3087:
3084:
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3073:
3069:
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3050:
3047:
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3037:
3033:
3029:
3026:
3024:
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3009:
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2998:
2994:
2990:
2987:
2984:
2981:
2976:
2968:
2965:
2962:
2956:
2954:
2947:
2943:
2931:
2930:
2929:
2906:
2903:
2898:
2896:
2889:
2885:
2877:
2873:
2866:
2863:
2860:
2854:
2852:
2845:
2841:
2829:
2828:
2827:
2824:
2822:
2796:
2793:
2787:
2782:
2778:
2774:
2771:
2768:
2765:
2760:
2752:
2749:
2746:
2740:
2738:
2726:
2723:
2717:
2712:
2708:
2704:
2701:
2698:
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2632:
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2602:
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2531:
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2486:
2480:
2477:
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2471:
2468:
2465:
2460:
2456:
2452:
2447:
2443:
2439:
2434:
2430:
2413:
2409:
2405:
2396:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2367:
2363:
2359:
2354:
2350:
2334:
2331:
2322:
2316:
2313:
2310:
2307:
2301:
2298:
2292:
2289:
2286:
2277:
2274:
2271:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2211:
2207:
2185:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2152:
2148:
2141:
2137:
2134:
2111:
2105:
2102:
2099:
2096:
2093:
2090:
2085:
2081:
2077:
2072:
2068:
2064:
2059:
2055:
2034:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2005:
2001:
1997:
1992:
1988:
1984:
1979:
1975:
1950:
1944:
1941:
1935:
1932:
1929:
1926:
1923:
1920:
1914:
1911:
1905:
1902:
1899:
1890:
1887:
1884:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1824:
1820:
1797:
1791:
1788:
1785:
1782:
1779:
1776:
1771:
1767:
1763:
1758:
1754:
1750:
1745:
1741:
1720:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1691:
1687:
1683:
1678:
1674:
1670:
1665:
1661:
1639:
1633:
1630:
1627:
1624:
1621:
1618:
1612:
1609:
1603:
1600:
1597:
1588:
1585:
1582:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
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1516:
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1500:
1496:
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1425:
1423:
1422:
1419:
1417:
1413:
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1402:
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1393:
1389:
1388:
1387:
1386:
1383:
1380:
1375:
1374:
1373:
1372:
1369:
1362:
1360:
1357:
1352:
1329:
1326:
1323:
1316:
1307:
1303:
1299:
1294:
1290:
1286:
1281:
1277:
1267:
1264:
1261:
1258:
1251:
1234:
1231:
1228:
1221:
1212:
1208:
1204:
1199:
1195:
1191:
1186:
1182:
1175:
1172:
1169:
1166:
1159:
1142:
1139:
1136:
1133:
1126:
1117:
1113:
1109:
1104:
1100:
1096:
1091:
1087:
1083:
1077:
1074:
1071:
1068:
1061:
1045:
1041:
1037:
1032:
1028:
1024:
1019:
1015:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
980:
964:
960:
956:
953:
950:
947:
944:
941:
938:
935:
930:
926:
922:
917:
913:
905:
904:
903:
901:
893:
891:
885:
875:
872:
868:
867:
866:
865:
864:
863:
862:
861:
852:
849:
845:
844:
843:
842:
841:
840:
839:
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829:
826:
821:
820:
819:
818:
817:
816:
815:
814:
807:
804:
800:
796:
792:
788:
787:
786:
785:
784:
783:
782:
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777:
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765:
761:
757:
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752:
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750:
745:
742:
738:
737:
736:
735:
732:
729:
724:
723:
722:
721:
718:
714:
706:
704:
703:
699:
695:
683:
682:
681:
677:
673:
655:
651:
647:
642:
638:
634:
629:
625:
616:
615:
614:
613:
610:
607:
606:Michael Hardy
603:
599:
595:
594:
593:
589:
585:
581:
580:70.107.165.59
577:
556:
552:
548:
543:
539:
535:
530:
526:
505:
502:
499:
496:
493:
490:
487:
484:
479:
475:
471:
466:
462:
458:
453:
449:
436:
434:
433:
429:
425:
424:67.255.12.129
420:
413:
411:
410:
406:
402:
398:
395:
392:
389:
374:
370:
366:
362:
361:
360:
359:
358:
357:
352:
349:
347:
345:
343:
320:
316:
312:
307:
303:
299:
294:
290:
282:
279:
275:
271:
267:
263:
262:
261:
260:
257:
253:
249:
245:
244:
243:
242:
238:
234:
216:
211:
207:
203:
198:
194:
190:
185:
181:
168:
153:
149:
148:High-priority
143:
140:
139:
136:
119:
115:
111:
110:
102:
96:
91:
89:
86:
82:
81:
77:
73:High‑priority
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
5711:— Preceding
5664:— Preceding
5623:— Preceding
5620:
5563:
5559:
5536:— Preceding
5533:
5489:69.159.12.95
5483:— Preceding
5479:
5438:69.159.12.95
5432:— Preceding
5428:
5425:
5421:
5403:
5384:— Preceding
5253:
5079:
4947:
4820:— Preceding
4816:
4797:
4776:
4772:
4768:
4764:
4760:
4756:
4752:
4748:
4746:
4654:
4640:
4598:
4576:
4568:
4314:
3629:
3605:
3601:
3597:
3579:
3560:
3334:
2927:
2825:
2818:
2559:
2556:
2550:
2546:
2543:
2540:
2537:
2212:
2208:
2153:
2149:
2146:
2135:
1825:
1821:
1523:
1512:
1487:
1476:
1466:Graphics Lab
1463:
1453:
1439:
1429:
1415:
1409:
1406:
1390:see above --
1368:Evil saltine
1364:
1353:
1350:
897:
889:
770:
710:
690:
601:
597:
440:
421:
417:
401:132.45.121.6
399:
396:
393:
390:
386:
341:
337:
277:
273:
269:
265:
172:
147:
107:
51:WikiProjects
34:
5456:applies. –
5429:- If c: -->
4747:applies if
4602:—Preceding
2568:NeilAFraser
2562:—Preceding
2516:Neil Parker
2510:—Preceding
713:dot product
574:—Preceding
123:Mathematics
114:mathematics
70:Mathematics
5786:Categories
5504:WP:SOFIXIT
5454:WP:SOFIXIT
3581:article.--
1356:— Sverdrup
772:Jmheneghan
756:Jmheneghan
5508:Help:Talk
4284:numbers."
2823:article.
1470:XcepticZP
694:Mason0190
672:Mason0190
342:WurmWoode
233:Mason0190
39:is rated
5713:unsigned
5689:triangle
5666:unsigned
5625:unsigned
5538:unsigned
5485:unsigned
5434:unsigned
5386:unsigned
4822:unsigned
4604:unsigned
3608:, where
2576:contribs
2564:unsigned
2524:contribs
2512:unsigned
1515:diagrams
1379:Delirium
900:triangle
728:Pfalstad
588:contribs
576:unsigned
4800:Danshil
1499:Page 21
1495:Page 20
1426:Rewrite
717:Orborde
711:In the
150:on the
41:C-class
5701:videos
5697:carbon
5653:videos
5649:carbon
5520:videos
5516:carbon
5466:videos
5462:carbon
4920:arccos
4886:arccos
4852:arccos
4575:Martin
4295:Ac44ck
3583:345Kai
3563:Ac44ck
1433:345Kai
1392:345Kai
1359:(talk)
1265:arccos
741:345Kai
276:&
266:obtuse
47:scale.
5685:Angle
5481:=b"
1412:roken
28:This
5765:talk
5761:Wqwt
5721:talk
5674:talk
5633:talk
5607:talk
5603:Wqwt
5590:talk
5546:talk
5493:talk
5442:talk
5409:talk
5394:talk
5371:5.25
5310:6.25
5304:5.25
5298:4.25
5214:6.25
5065:5.25
5004:6.25
4998:5.25
4992:4.25
4931:6.25
4928:4.25
4830:talk
4804:talk
4785:talk
4763:and
4647:talk
4612:talk
4590:talk
4299:talk
3587:talk
3567:talk
2928:And
2572:talk
2520:talk
1497:and
1418:egue
776:talk
760:talk
698:talk
676:talk
600:and
584:talk
428:talk
405:talk
369:talk
252:talk
237:talk
142:High
5565:Nik
5365:3.5
5359:1.5
5336:2.5
5277:3.5
5271:2.5
5265:1.5
5239:.25
5199:sin
5175:.25
5162:2.5
5148:sin
5124:.25
5108:2.5
5094:sin
5059:3.5
5053:1.5
5030:2.5
4971:3.5
4965:2.5
4959:1.5
4897:2.5
4863:2.5
4753:any
4751:is
4716:cos
4540:cos
4457:cos
4374:cos
4245:cos
4178:cos
4148:cos
4129:sin
4064:sin
4028:cos
4001:cos
3948:sin
3914:cos
3826:sin
3792:cos
3719:and
3675:sin
3658:cos
3524:cos
3477:cos
3416:cos
3335:Or
3298:cos
3245:cos
3181:cos
3134:cos
3070:cos
2995:cos
2779:sin
2709:cos
2654:cos
2478:cos
2388:cos
2314:cos
2177:cos
2103:cos
2026:cos
1942:180
1933:cos
1789:cos
1712:cos
1631:cos
1403:We?
1167:cos
1069:cos
1000:cos
948:cos
604:).
5788::
5767:)
5738:Δ
5723:)
5703:)
5699:•
5676:)
5655:)
5651:•
5635:)
5609:)
5592:)
5571:ai
5568:ol
5548:)
5522:)
5518:•
5495:)
5468:)
5464:•
5444:)
5411:)
5396:)
5362:×
5202:
5151:
5097:
5056:×
4923:
4889:
4855:
4832:)
4806:)
4787:)
4725:α
4719:
4704:−
4649:)
4631:--
4614:)
4592:)
4549:γ
4543:
4528:−
4466:β
4460:
4445:−
4383:α
4377:
4362:−
4301:)
4251:θ
4248:
4233:−
4184:θ
4181:
4166:−
4160:θ
4157:
4141:θ
4138:
4076:θ
4073:
4034:θ
4031:
4016:−
4013:θ
4010:
3954:θ
3951:
3923:−
3920:θ
3917:
3835:−
3832:θ
3829:
3801:−
3798:θ
3795:
3681:θ
3678:
3664:θ
3661:
3616:θ
3604:,
3600:,
3589:)
3569:)
3561:-
3533:θ
3527:
3511:−
3486:θ
3480:
3431:θ
3425:
3397:−
3307:θ
3301:
3282:−
3254:θ
3248:
3190:θ
3184:
3168:−
3143:θ
3137:
3085:θ
3079:
3048:−
3010:θ
3004:
2982:−
2794:θ
2788:
2750:−
2724:θ
2718:
2696:−
2660:θ
2657:
2642:−
2578:)
2574:•
2526:)
2522:•
2490:^
2481:
2466:−
2423:⇒
2406:−
2400:^
2391:
2360:−
2345:⇒
2332:−
2326:^
2317:
2275:−
2266:⇒
2253:×
2235:×
2206:.
2189:^
2180:
2115:^
2106:
2091:−
2048:⇒
2038:^
2029:
2011:−
1985:−
1970:⇒
1954:^
1945:−
1936:
1888:−
1879:⇒
1866:×
1848:×
1801:^
1792:
1777:−
1734:⇒
1724:^
1715:
1697:−
1671:−
1656:⇒
1643:^
1634:
1622:−
1586:−
1577:⇒
1564:×
1546:×
1377:--
1300:−
1268:
1205:−
1170:
1134:−
1097:−
1084:−
1072:
1025:−
1012:−
1003:
988:−
951:
936:−
801:|
797:|
793:|
778:)
762:)
700:)
678:)
590:)
586:•
571:?
485:−
430:)
407:)
371:)
254:)
239:)
5763:(
5747:C
5744:B
5741:A
5719:(
5695:(
5672:(
5647:(
5631:(
5605:(
5588:(
5574:h
5544:(
5514:(
5491:(
5460:(
5440:(
5407:(
5392:(
5368:=
5333:1
5307:,
5301:,
5274:,
5268:,
5234:5
5229:=
5222:2
5218:/
5210:1
5205:C
5170:5
5159:1
5154:B
5119:5
5114:=
5105:1
5100:A
5062:=
5027:1
5001:,
4995:,
4968:,
4962:,
4894:1
4860:1
4828:(
4802:(
4783:(
4777:a
4773:a
4769:a
4765:c
4761:b
4757:α
4749:a
4728:)
4722:(
4713:c
4710:b
4707:2
4699:2
4695:c
4691:+
4686:2
4682:b
4678:=
4673:2
4669:a
4645:(
4610:(
4588:(
4584:1
4581:5
4578:4
4552:)
4546:(
4537:b
4534:a
4531:2
4523:2
4519:b
4515:+
4510:2
4506:a
4502:=
4497:2
4493:c
4469:)
4463:(
4454:c
4451:a
4448:2
4440:2
4436:c
4432:+
4427:2
4423:a
4419:=
4414:2
4410:b
4386:)
4380:(
4371:c
4368:b
4365:2
4357:2
4353:c
4349:+
4344:2
4340:b
4336:=
4331:2
4327:a
4297:(
4254:.
4242:b
4239:a
4236:2
4228:2
4224:b
4220:+
4215:2
4211:a
4207:=
4196:2
4192:c
4175:b
4172:a
4169:2
4163:)
4152:2
4144:+
4133:2
4125:(
4120:2
4116:b
4112:+
4107:2
4103:a
4099:=
4088:2
4084:c
4068:2
4058:2
4054:b
4050:+
4045:2
4041:a
4037:+
4025:b
4022:a
4019:2
4005:2
3995:2
3991:b
3987:=
3976:2
3972:c
3962:2
3958:)
3945:b
3942:(
3939:+
3934:2
3930:)
3926:a
3911:b
3908:(
3905:=
3894:2
3890:c
3853:.
3846:2
3842:)
3838:0
3823:b
3820:(
3817:+
3812:2
3808:)
3804:a
3789:b
3786:(
3781:=
3778:c
3746:.
3743:)
3740:0
3737:,
3734:0
3731:(
3728:=
3725:C
3713:,
3710:)
3707:0
3704:,
3701:a
3698:(
3695:=
3692:B
3687:,
3684:)
3672:b
3667:,
3655:b
3652:(
3649:=
3646:A
3630:c
3606:c
3602:b
3598:a
3585:(
3565:(
3542:)
3536:2
3519:g
3515:m
3506:a
3502:m
3498:(
3495:)
3489:2
3472:g
3468:m
3464:+
3459:a
3455:m
3451:(
3448:=
3434:2
3420:2
3410:2
3405:g
3401:m
3392:2
3387:a
3383:m
3379:=
3370:2
3366:)
3359:2
3356:c
3350:(
3316:)
3310:2
3293:g
3289:m
3285:2
3277:a
3273:m
3269:2
3266:(
3263:)
3257:2
3240:g
3236:m
3232:2
3229:+
3224:a
3220:m
3216:2
3213:(
3210:=
3199:)
3193:2
3176:g
3172:m
3163:a
3159:m
3155:(
3152:)
3146:2
3129:g
3125:m
3121:+
3116:a
3112:m
3108:(
3105:4
3102:=
3088:2
3074:2
3064:2
3059:g
3055:m
3051:4
3043:2
3038:a
3034:m
3030:4
3027:=
3013:2
2999:2
2991:b
2988:a
2985:4
2977:2
2973:)
2969:b
2966:+
2963:a
2960:(
2957:=
2948:2
2944:c
2907:b
2904:a
2899:=
2890:g
2886:m
2878:2
2874:/
2870:)
2867:b
2864:+
2861:a
2858:(
2855:=
2846:a
2842:m
2797:2
2783:2
2775:b
2772:a
2769:4
2766:+
2761:2
2757:)
2753:b
2747:a
2744:(
2741:=
2727:2
2713:2
2705:b
2702:a
2699:4
2691:2
2687:)
2683:b
2680:+
2677:a
2674:(
2671:=
2651:b
2648:a
2645:2
2637:2
2633:b
2629:+
2624:2
2620:a
2616:=
2607:2
2603:c
2570:(
2518:(
2487:C
2475:b
2472:a
2469:2
2461:2
2457:a
2453:+
2448:2
2444:b
2440:=
2435:2
2431:c
2414:2
2410:a
2397:C
2385:.
2382:b
2379:a
2376:2
2373:=
2368:2
2364:c
2355:2
2351:b
2338:)
2335:a
2323:C
2311:b
2308:2
2305:(
2302:a
2299:=
2296:)
2293:c
2290:+
2287:b
2284:(
2281:)
2278:c
2272:b
2269:(
2259:F
2256:B
2250:C
2247:B
2244:=
2241:E
2238:B
2232:D
2229:B
2186:C
2174:b
2171:2
2168:=
2165:F
2162:C
2112:B
2100:c
2097:a
2094:2
2086:2
2082:c
2078:+
2073:2
2069:a
2065:=
2060:2
2056:b
2035:B
2023:.
2020:c
2017:a
2014:2
2006:2
2002:c
1998:=
1993:2
1989:a
1980:2
1976:b
1963:)
1960:)
1951:B
1939:(
1930:a
1927:2
1924:+
1921:c
1918:(
1915:c
1912:=
1909:)
1906:a
1903:+
1900:b
1897:(
1894:)
1891:a
1885:b
1882:(
1872:E
1869:A
1863:B
1860:A
1857:=
1854:F
1851:A
1845:D
1842:A
1798:B
1786:c
1783:a
1780:2
1772:2
1768:c
1764:+
1759:2
1755:a
1751:=
1746:2
1742:b
1721:B
1709:.
1706:c
1703:a
1700:2
1692:2
1688:c
1684:=
1679:2
1675:a
1666:2
1662:b
1649:)
1640:B
1628:a
1625:2
1619:c
1616:(
1613:c
1610:=
1607:)
1604:a
1601:+
1598:b
1595:(
1592:)
1589:a
1583:b
1580:(
1570:E
1567:A
1561:B
1558:A
1555:=
1552:F
1549:A
1543:D
1540:A
1416:S
1410:B
1336:]
1333:)
1330:b
1327:a
1324:2
1321:(
1317:/
1313:)
1308:2
1304:c
1295:2
1291:b
1287:+
1282:2
1278:a
1274:(
1271:[
1262:=
1259:C
1238:)
1235:b
1232:a
1229:2
1226:(
1222:/
1218:)
1213:2
1209:c
1200:2
1196:b
1192:+
1187:2
1183:a
1179:(
1176:=
1173:C
1146:)
1143:b
1140:a
1137:2
1131:(
1127:/
1123:)
1118:2
1114:c
1110:+
1105:2
1101:b
1092:2
1088:a
1081:(
1078:=
1075:C
1046:2
1042:c
1038:+
1033:2
1029:b
1020:2
1016:a
1009:=
1006:C
997:b
994:a
991:2
965:2
961:c
957:=
954:C
945:b
942:a
939:2
931:2
927:b
923:+
918:2
914:a
803:@
799:C
795:T
774:(
758:(
696:(
674:(
656:2
652:c
648:=
643:2
639:b
635:+
630:2
626:a
602:c
598:b
582:(
557:2
553:c
549:+
544:2
540:b
536:=
531:2
527:a
506:A
503:s
500:o
497:c
494:c
491:b
488:2
480:2
476:c
472:+
467:2
463:b
459:=
454:2
450:a
426:(
403:(
367:(
321:2
317:b
313:+
308:2
304:a
300:=
295:2
291:c
278:b
274:a
270:c
250:(
235:(
217:.
212:2
208:b
204:=
199:2
195:c
191:+
186:2
182:a
154:.
53::
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