From 79f30499252edc062616f00f2a2578b7c8397157 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Tue, 19 Mar 2019 11:38:17 +1100 Subject: [PATCH] [spec] arc[sin|cos|tan] --- spec/circ.md | 18 ++++++++++++++++++ 1 file changed, 18 insertions(+) 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. + +### $\arcsin$ + +$$\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y, \quad \text{where } \sin y = x \text{ and } y \in [{-\pi \over 2}, {\pi \over 2}]$$ + +### $\arcos$ + +$$\cos^{-1} \rightarrow \mathbb{R}, \quad \cos^{-1} x = y, \quad \text{where } \cos y = x \text{ and } y \in [0, \pi]$$ + +### $\arctan$ + +$$\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y, \quad \text{where } \tan y = x \text{ and } y \in \left(-{\pi \over 2}, {\pi \over 2}\right)$$ -- 2.43.2