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Applied Mathematics Cartesian Problems

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I have attached two problems (see attachment); I always have problems in solving the parts highlighted in red- part a and d. May I please have a detailed step-by-step method of finding the angle, area, and the Cartesian equation. On the last part, which is (d), I am always getting the signs wrong, so please show me the best method and how I can keep the signs.

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The solution uses applied mathematics to solve the Cartesian equations.

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