Diagram showing the relationship between arctangents and respective areas.<br><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\rm {area}}(PON)&amp;={\rm {area}}(MOF)=\pi \times {\frac {\angle MOF}{2\pi }}=\angle MEF=\arctan {1 \over 2}\\{\rm {area}}(POM)&amp;={\rm {area}}(NOF)=\arctan {1 \over 3}\\{\rm {area}}(POF)&amp;={\pi  \over 4}=\arctan {1 \over 2}+\arctan {1 \over 3}\\{\rm {area}}(MON)&amp;=\arctan {1 \over 7}\\\arctan {1 \over 2}&amp;=\arctan {1 \over 3}+\arctan {1 \over 7}\end{aligned}}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi></mrow></mrow><mo stretchy="false">(</mo><mi>P</mi><mi>O</mi><mi>N</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi></mrow></mrow><mo stretchy="false">(</mo><mi>M</mi><mi>O</mi><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>π<!-- π --></mi><mo>×<!-- × --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mi mathvariant="normal">∠<!-- ∠ --></mi><mi>M</mi><mi>O</mi><mi>F</mi></mrow><mrow><mn>2</mn><mi>π<!-- π --></mi></mrow></mfrac></mrow><mo>=</mo><mi mathvariant="normal">∠<!-- ∠ --></mi><mi>M</mi><mi>E</mi><mi>F</mi><mo>=</mo><mi>arctan</mi><mo>⁡<!-- ⁡ --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi></mrow></mrow><mo stretchy="false">(</mo><mi>P</mi><mi>O</mi><mi>M</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi></mrow></mrow><mo stretchy="false">(</mo><mi>N</mi><mi>O</mi><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>arctan</mi><mo>⁡<!-- ⁡ --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mtd></mtr><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi></mrow></mrow><mo stretchy="false">(</mo><mi>P</mi><mi>O</mi><mi>F</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>π<!-- π --></mi><mn>4</mn></mfrac></mrow><mo>=</mo><mi>arctan</mi><mo>⁡<!-- ⁡ --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mi>arctan</mi><mo>⁡<!-- ⁡ --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mtd></mtr><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi></mrow></mrow><mo stretchy="false">(</mo><mi>M</mi><mi>O</mi><mi>N</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mi>arctan</mi><mo>⁡<!-- ⁡ --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>7</mn></mfrac></mrow></mtd></mtr><mtr><mtd><mi>arctan</mi><mo>⁡<!-- ⁡ --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd><mi></mi><mo>=</mo><mi>arctan</mi><mo>⁡<!-- ⁡ --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>+</mo><mi>arctan</mi><mo>⁡<!-- ⁡ --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>7</mn></mfrac></mrow></mtd></mtr></mtable></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\rm {area}}(PON)&amp;={\rm {area}}(MOF)=\pi \times {\frac {\angle MOF}{2\pi }}=\angle MEF=\arctan {1 \over 2}\\{\rm {area}}(POM)&amp;={\rm {area}}(NOF)=\arctan {1 \over 3}\\{\rm {area}}(POF)&amp;={\pi  \over 4}=\arctan {1 \over 2}+\arctan {1 \over 3}\\{\rm {area}}(MON)&amp;=\arctan {1 \over 7}\\\arctan {1 \over 2}&amp;=\arctan {1 \over 3}+\arctan {1 \over 7}\end{aligned}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ab8f539780cf05b3d1b7965274173a2a622fe4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -13.005ex; width:66.668ex; height:27.176ex;" alt="{\begin{aligned}{{\rm {area}}}(PON)&amp;={{\rm {area}}}(MOF)=\pi \times {\frac  {\angle MOF}{2\pi }}=\angle MEF=\arctan {1 \over 2}\\{{\rm {area}}}(POM)&amp;={{\rm {area}}}(NOF)=\arctan {1 \over 3}\\{{\rm {area}}}(POF)&amp;={\pi  \over 4}=\arctan {1 \over 2}+\arctan {1 \over 3}\\{{\rm {area}}}(MON)&amp;=\arctan {1 \over 7}\\\arctan {1 \over 2}&amp;=\arctan {1 \over 3}+\arctan {1 \over 7}\end{aligned}}"></span>
<a href="//commons.wikimedia.org/wiki/User:Krishnavedala" title="User:Krishnavedala">Krishnavedala</a>
2014-08-27 14:36:21
<span class="int-own-work" lang="en">Own work</span>