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'derivation of half-angle formulae for tangent'
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| Title of object: |
derivation of half-angle formulae for tangent |
| Canonical Name: |
DerivationOfHalfAngleFormulaeForTangent |
| Type: |
Derivation |
| Created on: |
2007-04-27 23:57:46 |
| Modified on: |
2007-04-28 00:13:00 |
| Classification: |
msc:26A09 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
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%\usepackage{xypic}
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% define commands here
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Content:
Start with the angle duplication formula
\[
\tan (x) = {2 \tan (x/2) \over 1 - \tan^2 (x/2)}.
\]
Cross-multiply and move terms around:
\[
\tan (x) \tan^2 (x/2) + 2 \tan (x/2) = \tan (x)
\]
Divide by $\tan (x)$:
\[
\tan^2 (x/2) + {2 \tan (x/2) \over \tan x} = 1
\]
Add $1 / tan^2 (x)$ to both sides:
\[
\tan^2 (x/2) + {2 \tan (x/2) \over \tan x} + {1 \over \tan^2 (x)} = 1 + {1 \over \tan^2 (x)}
\]
\PMlinkname{Complete the square}{CompletingTheSquare}:
\[
\left (\tan (x/2) + {1 \over \tan (x)} \right)^2 = 1 + {1 \over \tan^2 (x)}
\]
Take a square root and move a term to obtain the half-angle formula:
\[
\tan (x/2) + \sqrt{ 1 + {1 \over \tan^2 (x)} } - {1 \over \tan (x)}
\] |
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