From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Tue, 19 Mar 2019 00:38:17 +0000 (+1100)
Subject: [spec] arc[sin|cos|tan]
X-Git-Tag: yr12~197
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/79f30499252edc062616f00f2a2578b7c8397157

[spec] arc[sin|cos|tan]
---

diff --git a/spec/circ.md b/spec/circ.md
index da26542..a32ec66 100644
--- a/spec/circ.md
+++ b/spec/circ.md
@@ -91,3 +91,21 @@ $$\sin 2x = 2 \sin x \cos x$$
 
 $$\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}$$
 
+## Inverse circular functions
+
+Inverse functions: $f(f^{-1}(x)) = x, \quad f(f^{-1}(x)) = x$  
+Must be 1:1 to find inverse (reflection in $y=x$
+
+Domain is restricted to make functions 1:1.
+
+### $\arcsin$
+
+$$\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y, \quad \text{where } \sin y = x \text{ and } y \in [{-\pi \over 2}, {\pi \over 2}]$$
+
+### $\arcos$
+
+$$\cos^{-1} \rightarrow \mathbb{R}, \quad \cos^{-1} x = y, \quad \text{where } \cos y = x \text{ and } y \in [0, \pi]$$
+
+### $\arctan$
+
+$$\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y, \quad \text{where } \tan y = x \text{ and } y \in \left(-{\pi \over 2}, {\pi \over 2}\right)$$