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# Pi/4=8arctan(1/10)-(rational fraction). What is the rational fraction? Thanks.

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Pi/4=8arctan(1/10)-(rational fraction). What is the rational fraction? Thanks.

Guest Aug 18, 2015

#3
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Reading this is like listening the same piece of music sung by two different singers. Except this is written down. Heurekas math is always like art too. It is like looking at the sheet music.

I not know how to read music but I can read this and maybe understand it if I study it long enough.

Dragonlance  Aug 19, 2015
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Since arctangent of 1/10 (in radians)=.09966865249........etc.  (1)

Then 8 x the above answer gives =.797349219929.....etc.        (2)

Then since 1/4 of Pi=Pi/4=.785398163......etc.                         (3)

Then subtract (2) from (3) above

Then=.0119510565......etc. And this is the arcrangent.             (4)

Then we find the tangent of (4) above

Which is:0.01195162554520335317930497716049....etc.

Guest Aug 19, 2015
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$$\small{\text{  \frac{\pi}{4}=8\cdot \arctan{(\frac{1}{10})}-(rational fraction). What is the rational fraction }} \\\\ \small{\text{ We start with:\qquad \tan{(\alpha)} = \frac{1}{10} }}\\\\ \small{\text{ Using the formula for double angles three times we get \tan{(8\alpha)} : }}\\\\ \small{\text{ \begin{array}{lrcl} \qquad \qquad &\tan{(2\alpha)} &=& \frac{2\cdot \tan{(\alpha)}} {1-\tan^2{(\alpha)}} = \frac{2\cdot \frac{1}{10} } {1- (\frac{1}{10})^2 } = \dfrac{20}{99} \\ \end{array}  }}\\\\ \small{\text{ Second Double angle formula, we get : }}\\\\ \small{\text{ \begin{array}{lrcl} \qquad \qquad &\tan{(4\alpha)} &=& \frac{2\cdot \tan{(2\alpha)}} {1-\tan^2{(2\alpha)}} = \frac{2\cdot \frac{20}{99} } {1- (\frac{20}{99})^2 } = \dfrac{3960}{9401} \\ \end{array}  }}\\\\ \small{\text{ Third Double angle formula, we get : }}\\\\ \small{\text{ \begin{array}{lrcl} \qquad \qquad &\tan{(8\alpha)} &=& \frac{2\cdot \tan{(4\alpha)}} {1-\tan^2{(4\alpha)}} = \frac{2\cdot \frac{3960}{9401} } {1- (\frac{3960}{9401})^2 } = \dfrac{74455920}{72697201} \\ \end{array}  }}\\\\ \small{\text{ 8\alpha differs from \frac{\pi}{4}, and \tan{( \frac{\pi}{4} )}=1  we have : }}\\\\ \small{\text{ \begin{array}{lrcl} \qquad \qquad &\tan{ (8\alpha-\frac{\pi}{4}) } &=& \frac{ \tan{(8\alpha)} - \tan{(\frac{\pi}{4})} } { \tan{(8\alpha)} + \tan{(\frac{\pi}{4})} } = \dfrac{ \frac{74455920}{72697201} - 1 } { \frac{74455920}{72697201} + 1 } = \dfrac{1758719}{147 153 121} \\ \end{array}  }}\\\\$$

$$\small{\text{ Taking the arctan of both sides, we have : }}\\\\ \small{\text{ \begin{array}{lrcl} \qquad \qquad & 8\alpha-\frac{\pi}{4}&=& \arctan{ (\frac{1758719}{147 153 121}) }\\\\ \qquad \qquad &\frac{\pi}{4}&=& 8\alpha - \arctan{ (\frac{1758719}{147 153 121}) } \\\\ \qquad \qquad &\mathbf{ \frac{\pi}{4} }& \mathbf{=} & \mathbf{ 8 \arctan{(\frac{1}{10})} - \arctan{ (\frac{1758719}{147 153 121}) } } \end{array}  }}$$

heureka  Aug 19, 2015
#3
+1311
+5