单位菱形面积定义正弦

单位菱形面积定义正弦 重建三角，全局皆活 2010年5月29日 发评论 阅读评论 本文标题引用了中国科学院院士张景中教授的文章《重建三角，全局皆活——初中数学课程结构性改革的一个建议》。今天无意间又找出这篇文章，读了个痛快～ 记得2007年5月份，我接到了一个任务，按张院士文章内容上一节课。要录像，要给张院士看。我欣然答应。 张院士的文章重建了三角函数的理论体系，三角函数的定义发生了根本的变化，“把边长为1，有一个角为A的菱形面积记作sinA”。由这个定义居然能推导出以下一些结论，有些内容连小学生都听得懂: 1、三角形面积 2、正弦的基本性质(1)sin0?=sin180?=0;(2)sin90?=1;(3)sinA=sin(180-A) 3、直角三角形中锐角的正弦 4、正弦定理 5、正弦和角公式 sin(α+β)=sinαsin(90?-β)+sinβsin(90?-α) 6、二倍角公式 sin2A=2sinAsin(90?-A) 7、特殊角的正弦值(1);(2);(3) 8、几何定理(1)直角三角形中，30?角的对边是斜边的一半;(2)勾股定理 9、正弦的增减性 10、三角形两边之和大于第三边 real, in each pen business lending Qian not for by pen query, led to mortgage failure of. (Vi) not in accordance with the relevant provisions of the Bank operation, failed to identify false documents, certificates, authorizations, warrants, etc. (VII) in the credit risk management system to provide false information, is not required by the guest rating credit management objective, impartial ratings of credit, resulting in credit decision-making errors. (H) the investigation report conclusions of false record, misleading statement or material omission, resulting in credit decision-making errors. (I) the orders, directives, orders, suggest that others not involved in the investigation business, issued by the investigators. (10) without the approval of authorities, after the first credit investigation, credit first and then record, counter-current process operations such as operation prior to approval. (11) article does not meet the risk management Department, Marketing Management Department to carry out supervision and inspection and due diligence evaluations or providing false information. (12) other violations result in credit business at substantial risk or significant potential risk behaviours. Sixth chapter supplementary articles article 28th of this approach by the Head Office of risk management Department is responsible for the interpretation and amendment. 29th spontaneous herein from the date of implementation. Qilu silver computer transmission of qilu Bank documents (2012), No. 289Qilu Bank personal credit on issuing the Notification Guide to business risk management branch, jurisdiction and the jurisdiction of business lines, Jinan branch, head office departments within its jurisdiction: in order to improve the quality of individual credit business, optimize the structure of assets, promotion of 这么多的内容，一节课里选什么内容好呢,我选了“正弦定义”、“三角形面积公式”、“正弦的性质”、“特殊角的正弦值”、“直角三角形中锐角的正弦”五个。当时上的班级是初一，结果因抽象程度太高，学生理解困难，练习题不会做。 我佩服张景中院士的智慧，谁能打破三角函数原有的体系,又有谁能建立新的体系?唯有景中～而且三角函数原有的性质、定理、公式一概不变，也就是“全局皆活”。但是，这样构建新体系好处在哪里,我不清楚。学生学习会遇到什么困难,我可以肯定学生遇到的困难比原来要多，低年级(如初一)的学生会有更多的困难。所以教材编写者不敢轻举妄动，还是袭用原来体系。 下面是我这节课的教案，发表于此，以飨读者。 一、认识正弦 同学们，我们都知道边长为1的正方形面积为1，如果把一个直角变为α，就成为一般的菱形，其面积就不是1了，究竟为多少呢,我们不知道，但肯定与α有关。我们不妨记作sinα。即一个角为α的菱形面积定义为sinα。 如果α=30?，面积=sin30?等。 当正方形ABCD的边长不是1时，设它的面积为S，变成一个角为α的菱形面积为s，我们可以证明s/S=sinα。 real, in each pen business lending Qian not for by pen query, led to mortgage failure of. (Vi) not in accordance with the relevant provisions of the Bank operation, failed to identify false documents, certificates, authorizations, warrants, etc. (VII) in the credit risk management system to provide false information, is not required by the guest rating credit management objective, impartial ratings of credit, resulting in credit decision-making errors. (H) the investigation report conclusions of false record, misleading statement or material omission, resulting in credit decision-making errors. (I) the orders, directives, orders, suggest that others not involved in the investigation business, issued by the investigators. (10) without the approval of authorities, after the first credit investigation, credit first and then record, counter-current process operations such as operation prior to approval. (11) article does not meet the risk management Department, Marketing Management Department to carry out supervision and inspection and due diligence evaluations or providing false information. (12) other violations result in credit business at substantial risk or significant potential risk behaviours. Sixth chapter supplementary articles article 28th of this approach by the Head Office of risk management Department is responsible for the interpretation and amendment. 29th spontaneous herein from the date of implementation. Qilu silver computer transmission of qilu Bank documents (2012), No. 289Qilu Bank personal credit on issuing the Notification Guide to business risk management branch, jurisdiction and the jurisdiction of business lines, Jinan branch, head office departments within its jurisdiction: in order to improve the quality of individual credit business, optimize the structure of assets, promotion of 上图中假设AB=3，则可以将正方形划分成9个单位正方形，菱形也如此，每个单位菱形与单位正方形的比都是sinα，所以s/S=sinα。 将上述问题中的正方形改为矩形，设矩形的面积为S，变成一个角为α的平行四边形面积为s，我们同样可以证明s/S=sinα。 这就说明将一个矩形的一个直角边成α后，面积是原来的sinα倍。张奠宙教授形象地说: 一个长方形(含正方形)，不改变其边长，当直角变为α时，面积所打的折扣是sinα。他将sinα比作“折扣”，非常形象非常贴切。从上图我们还可以看出，设AB=a，AD=b，?A=α，则。 二、 与正弦有关的面积公式 1、 边长分别为a、b，一个内角为α的平行四边形ABCD的面积是absinα; 2、 两边分别为a、b，夹角为α的三角形面积是. 三、 正弦的性质 认识了正弦之后，我们不难得到正弦的性质。当α=0?或180?时菱形的面积为0，所以sin0?=sin180?=0，这里列举性质如下: 1、sin0?=sin180?=0 2、sin90?=1 real, in each pen business lending Qian not for by pen query, led to mortgage failure of. (Vi) not in accordance with the relevant provisions of the Bank operation, failed to identify false documents, certificates, authorizations, warrants, etc. (VII) in the credit risk management system to provide false information, is not required by the guest rating credit management objective, impartial ratings of credit, resulting in credit decision-making errors. (H) the investigation report conclusions of false record, misleading statement or material omission, resulting in credit decision-making errors. (I) the orders, directives, orders, suggest that others not involved in the investigation business, issued by the investigators. (10) without the approval of authorities, after the first credit investigation, credit first and then record, counter-current process operations such as operation prior to approval. (11) article does not meet the risk management Department, Marketing Management Department to carry out supervision and inspection and due diligence evaluations or providing false information. (12) other violations result in credit business at substantial risk or significant potential risk behaviours. Sixth chapter supplementary articles article 28th of this approach by the Head Office of risk management Department is responsible for the interpretation and amendment. 29th spontaneous herein from the date of implementation. Qilu silver computer transmission of qilu Bank documents (2012), No. 289Qilu Bank personal credit on issuing the Notification Guide to business risk management branch, jurisdiction and the jurisdiction of business lines, Jinan branch, head office departments within its jurisdiction: in order to improve the quality of individual credit business, optimize the structure of assets, promotion of 3、sinα?=sin(180-α)? 4、 当α为锐角时，α越大sinα就越大;当α为钝角时，α越大sinα就越小。 练习一 1、 用计算器求值:sin30?，sin40?，sin45?，sin120?. 2、边长分别为2、4，一个内角为30度的平行四边形ABCD的面积是 。 3、两边分别为6、5，夹角为45度的三角形面积是 。 4、在括号内写出角的度数，使等式成立sin40?=sin( )，sin170?=sin( ). 四、正弦再认识 如图，在直角?ABC中，?C=90?，BC=a，AC=b，AB=c，那么有==，化简后得。 即 在直角三角形中，一个锐角的正弦等于这个角的对边与斜边之比。 练习二 1、如上图，?ABC中?C=90。，BC=6，sinA=0.6，求AB的长。 2、在第1题中如果AC=8，求sinB。 练习三 1、比较大小 (1)Sin30? sin80? (2)sin100? sin140? real, in each pen business lending Qian not for by pen query, led to mortgage failure of. (Vi) not in accordance with the relevant provisions of the Bank operation, failed to identify false documents, certificates, authorizations, warrants, etc. (VII) in the credit risk management system to provide false information, is not required by the guest rating credit management objective, impartial ratings of credit, resulting in credit decision-making errors. (H) the investigation report conclusions of false record, misleading statement or material omission, resulting in credit decision-making errors. (I) the orders, directives, orders, suggest that others not involved in the investigation business, issued by the investigators. (10) without the approval of authorities, after the first credit investigation, credit first and then record, counter-current process operations such as operation prior to approval. (11) article does not meet the risk management Department, Marketing Management Department to carry out supervision and inspection and due diligence evaluations or providing false information. (12) other violations result in credit business at substantial risk or significant potential risk behaviours. Sixth chapter supplementary articles article 28th of this approach by the Head Office of risk management Department is responsible for the interpretation and amendment. 29th spontaneous herein from the date of implementation. Qilu silver computer transmission of qilu Bank documents (2012), No. 289Qilu Bank personal credit on issuing the Notification Guide to business risk management branch, jurisdiction and the jurisdiction of business lines, Jinan branch, head office departments within its jurisdiction: in order to improve the quality of individual credit business, optimize the structure of assets, promotion of 2、二块全等的含有45?的三角板如图放置，则红色部分的两个三角形面积是否相等, 为什么, 相关文章: real, in each pen business lending Qian not for by pen query, led to mortgage failure of. (Vi) not in accordance with the relevant provisions of the Bank operation, failed to identify false documents, certificates, authorizations, warrants, etc. (VII) in the credit risk management system to provide false information, is not required by the guest rating credit management objective, impartial ratings of credit, resulting in credit decision-making errors. (H) the investigation report conclusions of false record, misleading statement or material omission, resulting in credit decision-making errors. (I) the orders, directives, orders, suggest that others not involved in the investigation business, issued by the investigators. (10) without the approval of authorities, after the first credit investigation, credit first and then record, counter-current process operations such as operation prior to approval. (11) article does not meet the risk management Department, Marketing Management Department to carry out supervision and inspection and due diligence evaluations or providing false information. (12) other violations result in credit business at substantial risk or significant potential risk behaviours. Sixth chapter supplementary articles article 28th of this approach by the Head Office of risk management Department is responsible for the interpretation and amendment. 29th spontaneous herein from the date of implementation. Qilu silver computer transmission of qilu Bank documents (2012), No. 289Qilu Bank personal credit on issuing the Notification Guide to business risk management branch, jurisdiction and the jurisdiction of business lines, Jinan branch, head office departments within its jurisdiction: in order to improve the quality of individual credit business, optimize the structure of assets, promotion of