2022年全国甲卷第16题的多解探究

Exploration of Multiple Solutions to Question 16 of the 2022 National Grade A Exam

摘要: 2022年全国甲卷16题主要是对高中知识余弦定理和求最值的知识点的考察，主要考察学科知识和学科素养。虽然难度中等，但此题题型和图像结构需要同学联系到解三角形的基本模型。余弦定理和几何知识本就联系紧密，这就使得本题可以从不懂的数学角度出现多种解法。同时在解三角形中要有方程思想、函数思想和不等式等思想。本文用波利亚解题思路引导进行一题多解，分析方法的优劣，提供在多解中灵活挑选最优解。

Abstract: This question is the last blank filling question in the 2022 national volume A and B, focusing on the examinees’ mastery of the Law of cosines, Pythagorean theorem, distance formula between two points, complementary Trigonometric functions formula, mean inequality, Discriminant, finding the maximum value of a function by using derivatives and other specific knowledge points, as well as the thought of combining logarithms with graphs, function thought understanding and applying mathematical concepts such as transformation of ideas. As the final question of filling in the blank, this question has a wide range of knowledge, low overall difficulty, and diverse methods. It has many flexible choices for students and requires high flexibility in their thinking. In terms of core competencies, this question examines aspects such as logical reasoning, computational ability, and geometric intuition.