Initial program 16.5
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
- Using strategy
rm Applied cos-diff3.6
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R\]
Taylor expanded around inf 3.6
\[\leadsto \color{blue}{\cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R\]
Simplified3.6
\[\leadsto \color{blue}{\cos^{-1} \left((\left((\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right))_*\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) + \left(\sin \phi_1 \cdot \sin \phi_2\right))_*\right)} \cdot R\]
- Using strategy
rm Applied add-cube-cbrt3.7
\[\leadsto \cos^{-1} \left((\left((\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right))_*\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) + \left(\sin \phi_1 \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \phi_2} \cdot \sqrt[3]{\sin \phi_2}\right) \cdot \sqrt[3]{\sin \phi_2}\right)}\right))_*\right) \cdot R\]
Applied associate-*r*3.7
\[\leadsto \cos^{-1} \left((\left((\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right))_*\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) + \color{blue}{\left(\left(\sin \phi_1 \cdot \left(\sqrt[3]{\sin \phi_2} \cdot \sqrt[3]{\sin \phi_2}\right)\right) \cdot \sqrt[3]{\sin \phi_2}\right)})_*\right) \cdot R\]
Final simplification3.7
\[\leadsto \cos^{-1} \left((\left((\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right))_*\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\sqrt[3]{\sin \phi_2} \cdot \left(\sin \phi_1 \cdot \left(\sqrt[3]{\sin \phi_2} \cdot \sqrt[3]{\sin \phi_2}\right)\right)\right))_*\right) \cdot R\]
