Know your continued fractions

Geometry Level 4

$8\sin { \frac { \pi }{ 10 } \cos { \frac { \pi }{ 5 } } } +4\sin { \frac { \pi }{ 10 } } +1=\alpha +\cfrac { 1 }{ \alpha +\cfrac { 1 }{ \alpha +\cfrac { 1 }{ \alpha +\ddots } } }$ Find $\alpha$ .

This problem is original.

Image credit: Wikimedia Levochik

The answer is 4.

1 solution

Rishabh Jain
Jan 4, 2016

$4\times\color{#3D99F6}{2 \sin \frac { \pi }{ 10 } \cos \frac { \pi }{ 5 }}+4\sin \frac { \pi }{ 10 } +1$ $=4(\sin\frac{3{\pi}}{10}-\sin\frac{\pi}{10})+4\sin \frac{\pi}{10}+1$ $=4\sin\frac{3\pi}{10}+1=4(\color{magenta}{\frac{\sqrt{5}+1}{4}})+1=\sqrt{5}+2$ $\text{Now}, \space \sqrt{5}+2=\alpha+\frac{1}{\sqrt{5}+2}$ $\alpha=(\sqrt{5}+2)-(\frac{1}{\sqrt{5}+2})$ $\color{#20A900}{\Rightarrow\alpha=4}\\$

Looks like your first latex soln.!Great job!!

Rohit Udaiwal - 5 years, 5 months ago

Learning to use latex.. :) . Are there any flaws in the solution that made you conclude this? (just asking)

Rishabh Jain - 5 years, 5 months ago

When writing text use this format \text{...} otherwise the text will appear in italics.For eg., $Hi~Rishabh$ (italics) and $\text{Hi Rishabh}$ .You can see the difference!Use \ before sin,and your solutions will become flawless!btw(+1)