2070: Trig Identities

2018-11-09T22:02:37Z

DaGriff: clarified the at^2=distance equation

{{comic
| number = 2070
| date = November 9, 2018
| title = Trig Identities
| image = trig_identities.png
| titletext = ARCTANGENT THETA = ENCHANT AT TARGET
}}

==Explanation==
{{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 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 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 subsequent lines then treat the trigonometric function as a product of variables rather than a function

The third line uses algebraic substitutions, since multiplication is commutative, such that:

<math>\frac{c}{s}\sin\theta=\frac{c}{s}\frac{b}{c}</math>

<math>\frac{a}{o}\cos\theta=\frac{a}{o}\frac{a}{c}</math>

<math>\frac{b}{n}\tan\theta=\frac{b}{a}\frac{b}{c}</math>

The fourth line continues the process using the line 2 COT identity to create words by multiplying both sides of the respective identities by:

<math>\frac{b}{c}</math> then by <math>\frac{a}{1}</math> then by <math>\frac{st}{b}</math>

The fifth line continues the 2-way interchangeability between algebraic variables and trig functions, where the expression is multiplied by <math>\frac{c}{c}</math>, which equals 1 (thus merely multiplying both sides of the equation by 1 and maintaining the equations value). Arranging the variables in a different order yields two of the original (and accurate) trig functions from lines 1 and 2

<math>\frac{b}{c}=\sin\theta</math> <math>\frac{c}{a}=sec\theta</math>

then substituting to yield

<math>\sin\theta\sec\theta</math>

or, rearranging the function characters as individual variables, should yield

<math>\mathrm{insect}\theta^2</math> .

Line 6 squares the TAN identity and expands the function then multiplies both sides by b.

Line 7 uses the expanded line 6 result brings in the physics of bananas via the kinematic formula for distance as a function of acceleration (a) and time (t).

Solving for <math>\mathrm{a}\mathrm{t}^2</math> yields:

<math>\mathrm{a}\mathrm{t}^2=\mathrm{distance}\frac{2}{1}</math> which simplifies to <math>\mathrm{distance2}</math>

Taking this equality and substituting into the line 6 expanded equation then further expanding gives us the final equation:

<math>\mathrm{distance2}\mathrm{banana}=\frac{b^3}{\theta^2}</math>

The slope of which is very, very slippery.

The title-text is an anagram resulting to a casting spell from common role playing games (ENCHANT AT TARGET would cast a spell on the target)

There seams to be at least one error or undiscovered algebraic substitution that turned the second S in line 5 to a T

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[Inside a single frame comic a right-angled triangle is shown. The short edges are labeled "a" and "b" respectively and the long edge has a "c". All angles are marked, the right angle by a square and the both others by an arc. One arc is labeled by the Greek symbol theta.]

:[Trigonometric functions on the marked angle theta in relation to "a", "b", and many more not depicted other variables are shown:]

:[Caption below the frame:]
:Key trigonometric identities

{{comic discussion}}

[[Category:Math]]

2070: Trig Identities

2018-11-09T21:59:35Z

DaGriff:

{{comic
| number = 2070
| date = November 9, 2018
| title = Trig Identities
| image = trig_identities.png
| titletext = ARCTANGENT THETA = ENCHANT AT TARGET
}}

==Explanation==
{{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 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 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 subsequent lines then treat the trigonometric function as a product of variables rather than a function

The third line uses algebraic substitutions, since multiplication is commutative, such that:

<math>\frac{c}{s}\sin\theta=\frac{c}{s}\frac{b}{c}</math>

<math>\frac{a}{o}\cos\theta=\frac{a}{o}\frac{a}{c}</math>

<math>\frac{b}{n}\tan\theta=\frac{b}{a}\frac{b}{c}</math>

The fourth line continues the process using the line 2 COT identity to create words by multiplying both sides of the respective identities by:

<math>\frac{b}{c}</math> then by <math>\frac{a}{1}</math> then by <math>\frac{st}{b}</math>

The fifth line continues the 2-way interchangeability between algebraic variables and trig functions, where the expression is multiplied by <math>\frac{c}{c}</math>, which equals 1 (thus merely multiplying both sides of the equation by 1 and maintaining the equations value). Arranging the variables in a different order yields two of the original (and accurate) trig functions from lines 1 and 2

<math>\frac{b}{c}=\sin\theta</math> <math>\frac{c}{a}=sec\theta</math>

then substituting to yield

<math>\sin\theta\sec\theta</math>

or, rearranging the function characters as individual variables, should yield

<math>\mathrm{insect}\theta^2</math> .

Line 6 squares the TAN identity and expands the function then multiplies both sides by b.

Line 7 uses the expanded line 6 result brings in the physics of bananas via the kinematic formula for distance as a function of acceleration (a) and time (t).

Solving for <math>\mathrm{a}\mathrm{t}^2</math> yields:

<math>\mathrm{distance}\frac{2}{1}</math> which simplifies to <math>\mathrm{distance2}</math>

Taking this equality and substituting into the line 6 expanded equation then further expanding gives us the final equation:

<math>\mathrm{distance2}\mathrm{banana}=\frac{b^3}{\theta^2}</math>

The slope of which is very, very slippery.

The title-text is an anagram resulting to a casting spell from common role playing games (ENCHANT AT TARGET would cast a spell on the target)

There seams to be at least one error or undiscovered algebraic substitution that turned the second S in line 5 to a T

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[Inside a single frame comic a right-angled triangle is shown. The short edges are labeled "a" and "b" respectively and the long edge has a "c". All angles are marked, the right angle by a square and the both others by an arc. One arc is labeled by the Greek symbol theta.]

:[Trigonometric functions on the marked angle theta in relation to "a", "b", and many more not depicted other variables are shown:]

:[Caption below the frame:]
:Key trigonometric identities

{{comic discussion}}

[[Category:Math]]

explain xkcd - User contributions [en]

2070: Trig Identities

2070: Trig Identities