feat: finish arcus functions

.husky/pre-commit

@@ -3,3 +3,4 @@
 
 npx pretty-quick --staged
 node .husky/scripts/cleanMaths.js
+git add .

languages/python/calculators/Trigometric Functions.py

@@ -1,10 +1,10 @@
-from cmath import e, sqrt, log
+from cmath import e, pi, sqrt, log
 from _helpers import floatInput
 
 i = sqrt(-1)
 
 compute = lambda numerator, denominator: (numerator / denominator).real
-ln = lambda x: log(x, e)
+computeInverse = lambda x: (-i * log(x, e)).real
 
 sin = lambda x: compute(pow(e, i * x) - pow(e, -i * x), 2 * i)
 cos = lambda x: compute(pow(e, i * x) + pow(e, -i * x), 2)
@@ -13,14 +13,20 @@ csc = lambda x: 1 / sin(x)
 sec = lambda x: 1 / cos(x)
 cot = lambda x: 1 / tan(x)
 
-arcsin = lambda x: (-i * ln((i * x) + sqrt(1 - pow(x, 2)))).real
+arcsin = lambda x: computeInverse((i * x) + sqrt(1 - pow(x, 2)))
+arccos = lambda x: computeInverse(x + sqrt(pow(x, 2) - 1))
+arctan = lambda x: computeInverse((i - x) / (i + x)) / 2
+arccsc = lambda x: computeInverse((pow(x, -1) * i) + sqrt(1 - pow(x, -2)))
+arcsec = lambda x: computeInverse((pow(x, -1)) + sqrt(pow(x, -2) - 1))
+arccot = lambda x: computeInverse((x + i) / (x - i)) / 2
 
-# todo: finish arc functions
+# todo: hyperbolic functions
-arccos = lambda x: None
+
-arctan = lambda x: None
+# todo: hyperbolic inverse functions
-arccsc = lambda x: (-i * (ln((pow(x, -1) * i) + sqrt(1 - pow(x, -2))))).real
+
-arcsec = lambda x: None
+# todo: hyperbolic reciprocal functions
-arccot = lambda x: None
+
+# todo: hyperbolic inverse reciprocal functions
 
 radians = floatInput("Please enter an amount of radians: ")
 

maths/style.cls

@@ -1,2 +1,8 @@
 \LoadClass[17pt]{extarticle}
 \pagenumbering{gobble}
+
+\newcommand{\euler}{\begin{gather*}
+	\text{By Euler's formula:} \\
+	e^{i\theta} = cos(\theta) + i\sin(\theta) \\
+	e^{-i\theta} = cos(\theta) - i\sin(\theta)
+\end{gather*}}

maths/trigometric functions/cos.tex

@@ -0,0 +1,27 @@
+\documentclass{../style}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\begin{document}
+\euler
+
+\begin{gather*}
+	\therefore \cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2}
+\end{gather*}
+
+\begin{gather*}
+	\text{let} \quad \cos(\theta) = x \\
+	2x = e^{i\theta} + e^{-i\theta} \\
+	2xe^{i\theta} = (e^{i\theta})^2 + 1 \\ 
+	(e^{i\theta})^2 + (-2x)e^{i\theta} + 1 = 0
+\end{gather*}
+
+\begin{gather*}
+	e^{i\theta} = \frac{-(-2x) \pm \sqrt{(-2x)^2 - 4}}{2} = x \pm \sqrt{x^2 - 1} \\
+	i\theta = \ln(x \pm \sqrt{x^2 - 1}) \\
+	\theta = -i\ln(x \pm \sqrt{x^2 - 1})
+\end{gather*}
+
+\begin{gather*}
+	\therefore \arccos(\theta) = -i\ln(\theta \pm \sqrt{\theta^2 - 1})
+\end{gather*}
+\end{document}

maths/trigometric functions/cot.tex

@@ -0,0 +1,29 @@
+\documentclass{../style}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\begin{document}
+\euler
+
+\begin{gather*}
+	\therefore \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
+	\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} \\
+	\tan(\theta) = \frac{\sin(\theta)}{\cos{\theta}} = -\frac{i(-1 + e^{2i\theta})}{1 + e^{2i\theta}} \\
+	\cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{1 + e^{2i\theta}}{i(-1 + e^{2i\theta})}
+\end{gather*}
+
+\begin{gather*}
+	\text{let} \quad \cot(\theta) = x \\
+	x(i + ie^{2i\theta}) = -1(1 + e^{2i\theta}) \\
+	ix + ixe^{2i\theta} = -e^{2i\theta} - 1 \\
+	ixe^{2i\theta} + e^{2i\theta} = -1 - ix \\
+	(ix + 1)e^{2i\theta} = -1 - ix \\
+	e^{2i\theta} = -\frac{1 - ix}{1 + ix} \\
+	2i\theta = \ln(\frac{x + i}{x - i}) \\
+	i\theta = \frac{1}{2}\ln(\frac{x + i}{x - i}) \\
+	\theta = -\frac{i}{2}\ln(\frac{x + i}{x - i})
+\end{gather*}
+
+\begin{gather*}
+	\therefore \text{arccot}(\theta) = -\frac{i}{2}\ln(\frac{\theta + i}{\theta - i})
+\end{gather*}
+\end{document}

maths/trigometric functions/csc.tex

@@ -2,11 +2,7 @@
 \usepackage{amsmath}
 \usepackage{amssymb}
 \begin{document}
-\begin{gather*}
+\euler
-	\text{By Euler's formula:} \\
-	e^{i\theta} = cos(\theta) + i\sin(\theta) \\
-	e^{-i\theta} = cos(\theta) - i\sin(\theta)
-\end{gather*}
 
 \begin{gather*}
 	\therefore \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
@@ -21,7 +17,7 @@
 \end{gather*}
 
 \begin{gather*}
-	e^{i\theta} = \frac{-(\frac{2i}{x}) \pm \sqrt{(\frac{2i}{x})^2 - 4(1)(-1)}}{2} = x^{-1}i \pm \sqrt{1 - x^2} \\
+	e^{i\theta} = \frac{-(\frac{2i}{x}) \pm \sqrt{(\frac{2i}{x})^2 - 4(-1)}}{2} = x^{-1}i \pm \sqrt{1 - x^2} \\
 	i\theta = \ln(x^{-1}i \pm \sqrt{1 - x^2}) \\
 	\theta = -i\ln(x^{-1}i \pm \sqrt{1 - x^2})
 \end{gather*}

maths/trigometric functions/sec.tex

@@ -0,0 +1,29 @@
+\documentclass{../style}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\begin{document}
+\euler
+
+\begin{gather*}
+	\therefore \cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} \\
+	\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{2}{e^{i\theta} + e^{-i\theta}}
+\end{gather*}
+
+\begin{gather*}
+	\text{let} \quad \sec(\theta) = x \\
+	x(e^{i\theta} + e^{-i\theta}) = 2 \\
+	e^{i\theta} + e^{-i\theta} = \frac{2}{x} \\
+	(e^{i\theta})^2 + 1 = \frac{2}{x}e^{i\theta} \\
+	(e^{i\theta})^2 + (-\frac{2}{x})e^{i\theta} + 1 = 0
+\end{gather*}
+
+\begin{gather*}
+	e^{i\theta} = \frac{-(-\frac{2}{x}) \pm \sqrt{(-\frac{2}{x})^2 - 4}}{2} = x^{-1} \pm \sqrt{x^{-2} - 1} \\
+	i\theta = \ln(x^{-1} \pm \sqrt{x^{-2} - 1}) \\
+	\theta = -i\ln(x^{-1} \pm \sqrt{x^{-2} - 1})
+\end{gather*}
+
+\begin{gather*}
+	\therefore \text{arcsec}(\theta) = -i\ln(\theta^{-1} \pm \sqrt{\theta^{-2} - 1})
+\end{gather*}
+\end{document}

maths/trigometric functions/sin.tex

@@ -2,11 +2,7 @@
 \usepackage{amsmath}
 \usepackage{amssymb}
 \begin{document}
-\begin{gather*}
+\euler
-	\text{By Euler's formula:} \\
-	e^{i\theta} = cos(\theta) + i\sin(\theta) \\
-	e^{-i\theta} = cos(\theta) - i\sin(\theta)
-\end{gather*}
 
 \begin{gather*}
 	\therefore \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i}
@@ -15,12 +11,12 @@
 \begin{gather*}
 	\text{let} \quad \sin(\theta) = x \\
 	2ix = e^{i\theta} - e^{-i\theta} \\
-	2ie^{i\theta}x = (e^{i\theta})^2 - 1 \\ 
+	2ixe^{i\theta} = (e^{i\theta})^2 - 1 \\ 
 	(e^{i\theta})^2 + (-2ix)e^{i\theta} - 1 = 0
 \end{gather*}
 
 \begin{gather*}
-	e^{i\theta} = \frac{-(-2ix) \pm \sqrt{(-2ix)^2 - 4(1)(-1)}}{2} = ix \pm \sqrt{1 - x^2} \\
+	e^{i\theta} = \frac{-(-2ix) \pm \sqrt{(-2ix)^2 - 4(-1)}}{2} = ix \pm \sqrt{1 - x^2} \\
 	i\theta = \ln(ix \pm \sqrt{1 - x^2}) \\
 	\theta = -i\ln(ix \pm \sqrt{1 - x^2})
 \end{gather*}

maths/trigometric functions/tan.tex

@@ -0,0 +1,28 @@
+\documentclass{../style}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\begin{document}
+\euler
+
+\begin{gather*}
+	\therefore \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
+	\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} \\
+	\tan(\theta) = \frac{\sin(\theta)}{\cos{\theta}} = -\frac{i(-1 + e^{2i\theta})}{1 + e^{2i\theta}}
+\end{gather*}
+
+\begin{gather*}
+	\text{let} \quad \tan(\theta) = x \\
+	x(1 + e^{2i\theta}) = -i(-1 + e^{2i\theta}) \\
+	x + xe^{2i\theta} = i - ie^{2i\theta} \\
+	xe^{2i\theta} + ie^{2i\theta} = i - x \\
+	e^{2i\theta}(i + x) = i - x \\
+	e^{2i\theta} = \frac{i - x}{i + x} \\
+	2i\theta = \ln(\frac{i - x}{i + x}) \\
+	i\theta = \frac{1}{2}\ln(\frac{i - x}{i + x}) \\
+	\theta = -\frac{i}{2}\ln(\frac{i - x}{i + x})
+\end{gather*}
+
+\begin{gather*}
+	\therefore \arctan(\theta) = -\frac{i}{2}\ln(\frac{i - \theta}{i + \theta})
+\end{gather*}
+\end{document}