feat: inverse csc calculator

.vscode/settings.json

@@ -35,6 +35,5 @@
         "streambuf": "cpp"
     },
     "latex-workshop.latex.autoBuild.run": "onSave",
-    "latex-workshop.latex.autoClean.run": "onBuilt",
-    "latex-workshop.cls"
+    "latex-workshop.latex.autoClean.run": "onBuilt"
 }

languages/python/calculators/Trigometric Functions.py

@@ -18,7 +18,7 @@ arcsin = lambda x: (-i * ln((i * x) + sqrt(1 - pow(x, 2)))).real
 # todo: finish arc functions
 arccos = lambda x: None
 arctan = lambda x: None
-arccsc = lambda x: None
+arccsc = lambda x: (-i * (ln((pow(x, -1) * i) + sqrt(1 - pow(x, -2))))).real
 arcsec = lambda x: None
 arccot = lambda x: None
 

maths/trigometric functions/csc.tex

@@ -0,0 +1,32 @@
+\documentclass{../style}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\begin{document}
+\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*}
+
+\begin{gather*}
+	\therefore \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
+	\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{2i}{e^{i\theta} - e^{-i\theta}}
+\end{gather*}
+
+\begin{gather*}
+	\text{let} \quad \csc(\theta) = x \\
+	\frac{2i}{x} = e^{i\theta} - e^{-i\theta} \\
+	\frac{2i}{x}e^{i\theta} = (e^{i\theta})^2 - 1 \\ 
+	(e^{i\theta})^2 + (-\frac{2i}{x})e^{i\theta} - 1 = 0
+\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} \\
+	i\theta = \ln(x^{-1}i \pm \sqrt{1 - x^2}) \\
+	\theta = -i\ln(x^{-1}i \pm \sqrt{1 - x^2})
+\end{gather*}
+
+\begin{gather*}
+	\therefore \text{arccsc}(\theta) = -i\ln(i\theta^{-1} \pm \sqrt{1 -\theta^2})
+\end{gather*}
+\end{document}