Editing 2070: Trig Identities
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==Explanation== | ==Explanation== | ||
− | This comic shows several real {{w|List_of_trigonometric_identities#Trigonometric_functions|trigonometric identities}} | + | {{incomplete|Please only mention here why this explanation isn't complete. Do NOT delete this tag too soon.}} |
+ | This comic shows several real and fictitious {{w|List_of_trigonometric_identities#Trigonometric_functions|trigonometric identities}}. Most of the identities past the second line are "derived" by applying algebraic methods to the letters in the trig functions, which violates the rules of math, since the trig functions are operators and not variables. | ||
− | The first line are | + | The first line are well known trigonometric functions. The second line contains the lesser known reciprocals of the trigonometric functions in the first line. |
− | The following identities are made up and are increasing in absurdity. The comic reflects on the confusion one gets when working more intensely with these identities, since there are a lot of hidden dependencies between them | + | The following identities are made up and are increasing in absurdity. The comic reflects on the confusion one gets when working more intensely with these identities, since there are a lot of hidden dependencies between them. |
− | The third and fourth line is made by treating the | + | The third and fourth line is made by treating the trigonometric function as a product of variables rather than a function and then using the above identities to create words. e.g. sin = b/c -> cin = b/s (this could also be a reference to the C++ cin). |
− | The | + | The third line is composed of puns: e.g., cin=b/s describes how the C++ cin object sucks, cas=0/c is read "California's South is Orange County". |
− | + | The second to last line performs some algebra on the individual letters of <math>(\mathrm{tan}\ \theta)^2=\frac{b^2}{a^2}</math> as a setup to the last line. The last line takes the formula <math>distance=\frac{1}{2}at^2</math> "from physics" and plugs it into the equation of the previous line, doing some algebra to replace <math>at^2</math> with <math>distance2</math> and expanding <math>(na)^2</math> into <math>nana</math> to get the final equation, <math>distance2banana=\frac{b^3}{\theta^2}</math> . This is valid algebra only if the trigonometric operators are taken as variable products rather than operators, but this is a common misconception encountered when people first learn trigonometry. The distance equation is the distance a constantly accelerating object initially at rest moves in a given length of time t, most often used to find how far an object dropped from rest will fall under the influence of gravity in a given amount of time (or how long it will take to fall a given distance). | |
− | * cas | + | There are a few formulas that may seem to have a mistake, but are in fact correct, just requiring more than one step: |
− | * In the identity sin | + | * <math>\mathrm{cas}\ \theta=\frac{o}{c}</math> seems to be derived from <math>\cos\theta=\frac{a}{c}</math> but to reach "cas" from "cos" one has to divide by "o" and multiply by "a". This would lead to <math>\frac{a^2}{co}</math> on the right hand side. However, this is still valid as it can be "proved" that both "a" and "o" are equal to 1, thus making <math>\frac{a^2}{co}=\frac{o}{c}</math> |
+ | * In the identity <math>\sin\theta\sec\theta=\mathrm{insect}\theta^2</math> one of the "s"'s has turned into a "t". This can be found by combining <math>\cos\theta=\frac{a}{c}</math>, <math>\mathrm{cas}\ \theta=\frac{o}{c}</math>, which show <math>o=a</math>, and <math>s=\frac{1}{c^2\theta}</math>. Using this with <math>\csc\theta=\frac{c}{b}</math> you can "prove" <math>c=b</math> and then with with <math>\cot\theta=\frac{a}{b}</math> you can find <math>t=\frac{1}{c^2\theta}=s</math>. The casual reader is more likely to see a 'poetic stretch' from the sound of saying 'sin sec' together. | ||
− | The title text is an | + | The title-text is an anagram. Due to the commutative property of multiplication (which states that order does not affect the product), these equations are equivalent if treated as individual variables as earlier. Another layer of absurdity is added in that the variable Theta is spelled out and broken into its letters, which are then treated as individual variables. (The {{w|arctangent}} referred to here is the inverse tangent, a one-sided inverse to the tangent function. You would not normally write <math>\arctan\theta</math>, since the theta in the comic refers to an angle, and the arctangent has an angle as its ''value'' rather than as its ''argument''; however, using theta here is merely unconventional, not forbidden.) The arctangent generally produces theta, the meaning of it being taken on theta being poorly understood. Randall here elucidates, via tongue-in-cheek algebraic proof, that taking a second arctangent of theta produces magical effects. |
− | === | + | ==== Value of Variables ==== |
− | + | It can be proven, given the six basic trig equations (the first two lines), under the (obviously false, but that's the point of the comic) assumption that each letter is a variable and they are being multiplied, that <b>all</b> the letters must be equal to 1. | |
+ | The proof can be conducted basically by setting things equal to each other and canceling/rearranging/replacing variables with what you discovered them to be equal to (like Randall did in lines 3-6), until a variable is proven to equal 1. Then that works its way around (with more setting things equal/rearranging/replacing) as every other variable is proven equal to the others, and to the one that equals 1. | ||
+ | (I would include the proof, but it is long and annoying to write. Sorry) | ||
− | + | Since every variable equals 1, all combinations of the letters a, b, c, e, i, n, o, s, t, and <math>\theta</math> can be validly set equal to each other. For example, biostatisticians=nonscientists, tobacco=assassinates, and even teta=<math>\theta</math>. Of course, it would be a massive pain to derive those as Randall derived the others, but by proving they all equal one, we know it can be done. | |
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==Transcript== | ==Transcript== | ||
+ | {{incomplete transcript|Do NOT delete this tag too soon.}} | ||
:[Inside a single frame comic a right-angled triangle is shown. The shorter sides are labeled "a" and "b" and the hypotenuse has a "c". All angles are marked: the right angle by a square and the two others by arcs. One arc (enclosed by "a" and "c") is labeled by the Greek symbol theta (θ).] | :[Inside a single frame comic a right-angled triangle is shown. The shorter sides are labeled "a" and "b" and the hypotenuse has a "c". All angles are marked: the right angle by a square and the two others by arcs. One arc (enclosed by "a" and "c") is labeled by the Greek symbol theta (θ).] | ||