2015 SS 28

2015 SS 28

1.

cos-theorem:

\(\begin{array}{|rcll|} \hline BM^2&=&x^2+\left(\dfrac{x}{2}\right)^2-2\cdot x \cdot \dfrac{x}{2}\cos(108^\circ) \\ BM^2 &=& x^2\left(1+\dfrac{1}{4}-\cos(108^\circ)\right) \quad |\quad \cos(108^\circ)=\cos(180^\circ-72^\circ)=-\cos(72^\circ) \\ BM^2 &=& x^2\left( \dfrac{5}{4}+\cos(72^\circ)\right) \\ \mathbf{BM} &\mathbf{=}& \mathbf{x \sqrt{\dfrac{5}{4}+\cos(72^\circ)}} \\ \hline \end{array} \)

2.
sin-theorem:

\(\begin{array}{|rcll|} \hline \dfrac{\sin(z)}{x} &=& \dfrac{\sin(36^\circ)}{ZB} \\\\ \mathbf{ZB} &\mathbf{=}& \mathbf{\dfrac{\sin(36^\circ)}{\sin(z)}x }\\ \hline \end{array} \)

3.

sin-theorem:

\(\begin{array}{|rcll|} \hline \dfrac{\sin(180^\circ-z)}{\frac{x}{2}} &=& \dfrac{\sin(72^\circ)}{ZM} \\\\ \dfrac{2\sin(z)}{x} &=& \dfrac{\sin(72^\circ)}{ZM} \\\\ \mathbf{ZM} &\mathbf{=}& \mathbf{\dfrac{\sin(72^\circ)}{2\sin(z)}x }\\ \hline \end{array} \)

\(\mathbf{z=\ ?}\)

\(\begin{array}{|rcll|} \hline BM &=& ZB+ZM \\ BM&=& \dfrac{\sin(36^\circ)}{\sin(z)}x+\dfrac{\sin(72^\circ)}{2\sin(z)}x \\\\ BM&=& \dfrac{x}{\sin(z)} \left( \sin(36^\circ) + \dfrac{\sin(72^\circ)}{2}\right) \quad | \quad BM=x \sqrt{\dfrac{5}{4}+\cos(72^\circ)} \\\\ x \sqrt{\dfrac{5}{4}+\cos(72^\circ)}&=& \dfrac{x}{\sin(z)} \left( \sin(36^\circ) + \dfrac{\sin(72^\circ)}{2}\right) \\\\ \sqrt{\dfrac{5}{4}+\cos(72^\circ)}&=& \dfrac{1}{\sin(z)} \left( \sin(36^\circ) + \dfrac{\sin(72^\circ)}{2}\right) \\\\ \mathbf{\sin(z)} & \mathbf{=} & \mathbf{ \dfrac{ \sin(36^\circ) + \dfrac{\sin(72^\circ)}{2} }{\sqrt{\dfrac{5}{4}+\cos(72^\circ)}} } \\\\ z &=& 180^\circ- \arcsin\left( \dfrac{ \sin(36^\circ) + \dfrac{\sin(72^\circ)}{2} }{\sqrt{\dfrac{5}{4}+\cos(72^\circ)}}\right) \\ &=& 180^\circ- \arcsin\left( \dfrac{ 1.06331351044 }{1.24860602048} \right) \\ &=& 180^\circ- \arcsin\left( 0.85160049928 \right) \\ &=& 180^\circ- 58.3861775592^\circ \\ \mathbf{z} &\mathbf{=}& \mathbf{121.613822441^\circ} \\ \hline \end{array}\)

\(\text{Let $\angle ABM =180^\circ-(36^\circ+z) $} \\ \text{Let $ 36^\circ+z = 36^\circ+ 121.613822441^\circ=157.613822441^\circ$} \)

4.
sin-theorem:

\(\begin{array}{|rcll|} \hline \dfrac{\sin(z)}{x} &=& \dfrac{\sin(ABM)}{3} \quad | \quad \angle ABM =180^\circ-(36^\circ+z) \\\\ \dfrac{\sin(z)}{x} &=& \dfrac{\sin\Big(180^\circ-(36^\circ+z)\Big)}{3} \\\\ \dfrac{\sin(z)}{x} &=& \dfrac{\sin( 36^\circ+z )}{3} \\\\ \mathbf{x} &\mathbf{=}& \mathbf{3\cdot \dfrac{\sin(z)}{\sin( 36^\circ+z )} } \\\\ x & = & 3\cdot \dfrac{0.85160049928}{\sin( 157.613822441^\circ )} \\\\ x & = & 3\cdot \dfrac{0.85160049928}{0.38084732121} \\\\ x & = & 3\cdot 2.23606797751 \quad | \quad 2.23606797751 = \sqrt{5} \\ \mathbf{x} &\mathbf{=}& \mathbf{3\cdot \sqrt{5}} \\ \hline \end{array}\)