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question:Determine the following points of a triangle given by its side-angle-side configuration: sides 11 and 6, and an included angle of frac{pi}{9}. The points to find are the centroid, circumcenter, incenter, orthocenter, nine-point center, and symmedian point.
answer:Circumcenter: left{frac{1}{2} sqrt{157-132 cos left(frac{pi }{9}right)},frac{1}{2} sqrt{157-132 cos left(frac{pi }{9}right)} cot left(frac{pi }{9}right)right} Centroid: left{frac{193-198 cos left(frac{pi }{9}right)}{3 sqrt{157-132 cos left(frac{pi }{9}right)}},frac{22 sin left(frac{pi }{9}right)}{sqrt{157-132 cos left(frac{pi }{9}right)}}right} Orthocenter: left{frac{36-66 cos left(frac{pi }{9}right)}{sqrt{157-132 cos left(frac{pi }{9}right)}},frac{left(6 cos left(frac{pi }{9}right)-11right) left(11 cos left(frac{pi }{9}right)-6right) csc left(frac{pi }{9}right)}{sqrt{157-132 cos left(frac{pi }{9}right)}}right} SymmedianPoint: left{frac{3 sqrt{157-132 cos left(frac{pi }{9}right)} left(11 cos left(frac{pi }{9}right)-12right)}{66 cos left(frac{pi }{9}right)-157},frac{33 sin left(frac{pi }{9}right) left(157-132 cos left(frac{pi }{9}right)right)^{3/2}}{29005-31086 cos left(frac{pi }{9}right)+4356 cos left(frac{2 pi }{9}right)}right} NinePointCenter: left{frac{229-264 cos left(frac{pi }{9}right)}{4 sqrt{157-132 cos left(frac{pi }{9}right)}},-frac{left(157 cos left(frac{pi }{9}right)-132right) csc left(frac{pi }{9}right)}{4 sqrt{157-132 cos left(frac{pi }{9}right)}}right} Incenter: left{frac{6 left(6+sqrt{157-132 cos left(frac{pi }{9}right)}-11 cos left(frac{pi }{9}right)right)}{17+sqrt{157-132 cos left(frac{pi }{9}right)}},frac{66 sin left(frac{pi }{9}right)}{17+sqrt{157-132 cos left(frac{pi }{9}right)}}right}
question:In the proof of the countability of the product Z_+ times Z_+, what is the significance of the subset A and the function g(x, y)?
answer:The subset A represents the set of pairs (x, y) where y le x, which forms a "lower triangular" region in the product Z_+ times Z_+. The function g(x, y) maps each pair in A to a unique positive integer. This mapping is crucial because it establishes a bijection between A and Z_+, which in turn implies that the product Z_+ times Z_+ is countable.
question:Which war in the mid-19th century weakened the economy of the Ottoman Empire?
answer:Crimean War Explanation: The Crimean War, fought from 1853 to 1856, was a major conflict between the Russian Empire and an alliance of the Ottoman Empire, France, the United Kingdom, and Sardinia. The war was fought primarily in the Crimean Peninsula, and it resulted in a significant loss of life and resources for all sides involved. The Ottoman Empire, in particular, suffered severe economic damage as a result of the war, as it was forced to borrow heavily from foreign banks to finance its war effort. The war also led to a decline in trade and commerce, as well as a sharp increase in inflation. The economic consequences of the Crimean War contributed to the decline of the Ottoman Empire in the late 19th and early 20th centuries.
question:Find the reduced row echelon form of the following matrix: left( begin{array}{cccc} 2 & 4 & -2 & 6 0 & 2 & 1 & -2 1 & 3 & -1 & 4 end{array} right).
answer:left( begin{array}{cccc} 1 & 0 & 0 & 1 0 & 1 & 0 & -1 0 & 0 & 1 & -1 end{array} right)