Trigonometry
Reciprocal identities
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
Secant and cosecant not taught in Sweden.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
If you prefer expressing them in sine and cosine:
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
Interestingly, note that tan is 1/cotan, but sine is not 1/cosine.
Offset identities
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
Imagine the unit circle. Instead of starting at zero, you start at pi/2. Then, due to subtraction, you walk clockwise, not counterclockwise.
Unit circle
Inverse trig functions
There are different definitions of the domain of inverse trig functions. Multi-valued arcsin is a thing. Mathematica in particular might define the cotangent differently from Sweden. But here are the rules for basic-uni Swedish math.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
TODO Explain how to think, don't just make a reference table
Law of sines
We can use it to find stuff about nonright triangles.
Angle A, opposing side a;
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
Num and denom can be swapped.
Law of cosines
Can tell you stuff about nonright triangles. Like with the Law of Sines, the capital letters denote angle opposing a side denoted in lower case.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
The Pythagorean Theorem is a special case of this Law.
Pythagorean identity
In Swedish, it's called trigonometriska ettan. cos2x + sin2x = 12
Should be obvious if you look at the unit circle and remember the Pythagorean Theorem.
Half-angle formula
Double-angle formula
sin(2x) = 2 sin(x) cos(x)
Proof?
Example: Find the sine of π/12
Using the double angle formula, we know that
sin(π/6) = 2 sin(π/12) cos(π/12).
We also know from our standard angles that sin(π/6) is 1/2. So
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
The reason I left cosine by itself is that we'll exploit the Pythagorean identity. First, let's square this thing.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
Now, look at a different equation, the Pythagorean identity:
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
We have an expression for node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
Multiply the equation so we're rid of the fraction.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
Now, we can do variable substitution. Say node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
Solve like any second-degree equation.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
So, given that we made y the squared sine of π/12, we square root the equation and get our answer.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
To figure out whether the plus or minus applies, test them. The result node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
Euler's formula🔗
General form: eiθ = cos(θ) + isin(θ)
Insert π and check the value. It gives you zero sin, -1 cos. So: eiπ = -1. You quickly realize that inputting any multiple of pi will give you a real number.
Trig angle-addition formulas can be derived: substitute (a + b) for θ. Convert to cis(a + b) form. Perform the complex multiplication. The terms with i will correspond to the angle addition formula for sin, since the sine axis in the unit circle and the imaginary axis are both vertical. The terms without i then, of course, correspond to the angle addition formula for cos.
De Moivre's formula follows from Euler's formula. (cosθ + isinθ)n is (eiθ)n, which by simple exponent rules is eiθn, which by definition is cos(nθ) + isin(nθ).
De Moivre's formula🔗
(cosθ + isinθ)n is equal to cos(nθ) + isin(nθ). You can state the cisθ in terms of Euler's formula to prove it.
Angle addition
To get the formula for tan, merely do the angle addition for sin, divided by angle addition for cos. Tan is sin/cos.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
Multiply both numerator and denominator by node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
node:internal/modules/cjs/loader:1228
throw err;
^
Error: Cannot find module 'katex'
Require stack:
- /home/kept/private-dotfiles/.config/emacs/texToMathML.js
at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15)
at Module._load (node:internal/modules/cjs/loader:1051:27)
at Module.require (node:internal/modules/cjs/loader:1311:19)
at require (node:internal/modules/helpers:179:18)
at Object.
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