2070: Trig Identities
Title text: ARCTANGENT THETA = ENCHANT AT TARGET
This comic shows several real trigonometric identities at the first two lines and further below some identities "derived" by applying algebraic methods to the letters in the trigonometric function names, which is obviously nonsense.
The first line are the known trigonometric functions: sine, cosine and tangent, and the second line contains the reciprocals of the trigonometric functions from the first line: cosecant, secant, and cotangent.
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. You can also check how they are related through the various Trigonometry Formulas.
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 second to last line performs some algebra on the individual letters of (tan θ)² = b²/a² as a setup to the last line. The last line takes the formula distance = 1/2 at² "from physics" and plugs it into the equation of the previous line, doing some algebra to replace at² with distance2 and expanding (na)² into nana to get the final equation, distance2banana = b³/θ². 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).
There are a few formulas that have mistakes if you simply make algebraic manipulations to the six standard trigonometric functions.
- cas θ = o/c seems to be derived from cos θ = a/c but to reach "cas" from "cos" one has to divide by "o" and multiply by "a". This would lead to cas θ = a²/oc.
- In the identity sin θ sec θ = insect θ² one of the "s"'s has turned into a "t", however this may be reached by 'phonetic stretch' from the sound of saying 'sin sec' together being similar to the sound of the word "insect". Another possible conversion is if you treat "s" as seconds, then "t" could be time, which keeps with the identity theme.
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 arctangent referred to here is the inverse tangent, a one-sided inverse to the tangent function. You would not normally write , 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.
From physics (and beyond)
The formula s=1/2 at² gives the distance a uniform accelerating object reaches over time. The second formula belongs to astronomy and the third law of Kepler in which the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit, meaning the fraction of b3 and t2 is a constant (banana).
But using the angle θ as an argument leads to Richard Feynman, who did many famous Lectures on Physics and his lost lecture about the Motion of Planets Around the Sun from 1964 in which he only used geometry, based on the orbital ellipse, a circle around, and matching right-angled triangles to illustrate this law from Kepler. For deeper understanding why it really does work there is a nice presentation at the "Journal of Symbolic Geometry": Feynman Says: “Newton implies Kepler, No Calculus Needed! (Brian Beckman, 2006)”
Proof of algebraic mistakes in the comic
Some have tried to argue there are mathematical justifications for the errors in some of the formulas, by stating (without proof) that you could prove that valid solutions to the original six trig identities (where letters are taken to be variables multiplied together) can be manipulated to show that solutions must have
- a=o and s=t.
These proofs are incorrect and can be shown easily with a counterexample. If you make the following assignments of variables like
- o=s=1/2 and set c=e=2
- while leaving the other variables set to 1 (a=b=i=n=t=θ=1). This variable assignment will simultaneously satisfy all six original trig identities:
However in this valid assignment, we have
- and we have as .
This demonstrates that you can not make a valid algebraic derivation of
without additional assumptions beyond the six given trigonometric identities.
- [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).]
- [Supposed trigonometric functions of the marked angle θ are shown:]
- sin θ = b/c
- cos θ = a/c
- tan θ = b/a
- cot θ = a/b
- sec θ = c/a
- csc θ = c/b
- cin θ = b/s
- cas θ = o/c
- tab θ = b²/na
- bot θ = a/c → boat θ = a²/c → stoat θ = a²/c · st/b
- tan θ ( = b/a = b/a · c/c = b/c · c/a = sin θ sec θ ) = insect θ²
- (tan θ)² = b²/a² ( → t²n²a⁴ = b²/θ² → at²ba(na)² = b³/θ²
- from physics: distance = 1/2 at² → ) distance2banana = b³/θ²
- [Caption below the frame:]
- Key trigonometric identities
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