\( \left. \begin{array} { l } { 10 ^ { 2 } = ( 13 - 15 ) ^ { 2 } + ( 13 - 5 ) ^ { 2 } - 2 ( 8 ) ^ { 2 } \cos \angle ( 0 ) } \\ { \cos \angle C O D ) } \end{array} \right. \)

To solve the equation \( 10^2 = (13 - 15)^2 + (13 - 5)^2 - 2(8)^2 \cos \angle (0) \), we first simplify the terms. We have \(10^2 = 100\), \( (13 - 15)^2 = 4 \), and \( (13 - 5)^2 = 64 \). Plugging these back into the equation leads to \( 100 = 4 + 64 - 128 \cos(0) \). This further simplifies to \( 100 = 68 - 128 \cos(0) \). Now, isolating \(\cos(0)\), we need to rearrange it to find its value. The relationship \( \cos(0) = 1 \) means \(100 = 68 - 128\), leading us to the significance of angles—always keep your results checked against core principles like the spherical and plane geometric properties in mathematics! One way to apply this concept in real life is in navigation and surveying. For instance, land surveyors use angles and trigonometric functions to establish precise locations and distances. By using the law of cosines, they can measure angles between points and determine unknown distances, which is crucial in planning and construction projects. So, the next time you're looking at a map, remember the hidden geometric connections!