Research on Mathematics Education

Research on Mathematics Education
Final Project
A research on case studies on the use of video-based instruction for the teaching and learning of mathematics.
ED359
Yeong Haur Kok

Abstract The purpose of this report is to give an overview on the issues pertaining to the use of videobased instruction for the teaching and learning of mathematics. This paper examined the existing literature and several projects to highlight the reasons and potential for the use of videos in math, and the important issues when designing such instruction for use in the classroom. The findings of this research indicate the use of video-based instruction has positive influences on the students in terms of performances and attitude and motivation. The report concludes with two recommendations for further research: the Introduction The potential of video as an educational tool has been envisioned by many even in the very early stages of motion picture. Thomas A, Edison saw the motion picture as being destined to revolutionize the educational system and in his own words “…in a few years it will supplant largely, if not entirely, the use of textbooks” Videos provide a highly efficient way of conveying information. If one picture is worth a thousand words, a 20-minute video must be worth millions of words. Visual images make a much greater impact than printed or spoken words. People tend to forget works they hear or read, but images are retained for a long time because they have emotional as well as intellectual appeal (Apostol, 1993). Video images handle mathematical concepts in a way that cannot be done only through text or still images as movement and sound introduce many components that help the learning process. Many of the developments on computer graphics, visualization and on mathematical software are used to video material with mathematical content. These provided opportunities to design and deliver a rich learning experience and indeed, changes have already taken place in the use of video-based instruction for the teaching and learning of mathematics. Current Situation Currently, the use of video for mathematics is mainly focused on teachers. There are several video-based learning and resources designed for teachers. For example, PBS TeacherSource is an online resource for teachers where they can search for videos of teachers modeling math lessons. The teacher can search by grade (K-12) and topic, and each video comes with a title and description, as well as the lesson plans detailing the objectives, activities and materials required. As the focus of this research is on video-based instruction for the teaching and learning of mathematics for students, this paper will not review the use of videos in teacher training or videos as resources for teachers.
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Literature Review A list of the papers reviewed for this research study is given in the references at the end of the paper. A total of nine papers were reviewed; Norma’s article “Things That Make Us Smart” were quoted but not formally reviewed for the purpose of this paper. Case Studies For this research, the following case studies were reviewed: ? Project Mathematics!
Project Mathematics! is a video-based instructional project designed for the teaching and learning mathematics. It was launched by Tom Apostol and Jim Blinn from the California Institute of Technology in 1987. The goal of the project was to attract young people to mathematics through high-quality instructional videos that show mathematics to be understandable, exciting, and with several applications to daily life. By the year 2000, the project had produced ten broadcast quality videotapes used as support material in high school and community college classrooms. Each module consists of a videotape, not exceeding 30 minutes in length, and a workbook/study guide that elaborates on the important ideas in the video and provides exercise for the students. The workbook is written by Tom Apostol and is divided into sections, the same way the video is. In each section of the workbook, the author summarized the important ideas of the corresponding video section and also provided additional information that enlarges the concepts transmitted on the video. The titles of the programs produced to date are: o o o o o o o o The Theorem of Pythagoras The Story of Pi Similarity Polynomials Teachers Workshop Sines and Cosines Part I, II, III The Tunnel of Samos Early History of Mathematics
More than 140,000 copies of Project Mathematics! videotapes were in circulation by year 2000. The modules won a dozen prestigious awards for excellence in educational video production and the materials were distributed in Australia, Canada, Denmark and Great Britain. Eight modules were translated into Hebrew for broadcast on Israeli Educational Television, and
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also in Portuguese by the University of Lisbon Center of Mathematics and Fundamental Applications. ? Touching Soap Films
This is a popular educational video on minimal surfaces. A soap film is physically similar to a piece of rubber surface which tries to contract itself under surface tension to a surface with least area. Surfaces with least area appear as optimal solutions of many problems. The video uses an animated character Kalle and through his exploration of the palace of soap films, introduces and explains the concepts of minimal surfaces. All animations are computer generated. Behind each aesthetically pleasing shape and object are mathematical concepts, as well as real-world applications in architecture, chemistry and physics. However, while the video is good and interesting, it is quite difficult to use as an educational device without supplemental explanation. Fortunately, each package comes with a booklet explaining the concepts featured in the video, making it possible and easy to incorporate into courses on applied math. Besides the American version, there are English, German and Portuguese versions as well. ? Encore’s Vacation (Taiwan)
“Encore’s Vacation” is a Taiwanese video series based on “The Jasper Series”, a program of anchored instruction developed by the Cognition and Technology Group at Vanderbilt (CTGV) to teach mathematics. “The Jasper Series” full name is called “The Adventure of Jasper Woodbury Mathematical Problem Solving series (The Jasper Series)”. It is based on a set of theory-based design principles, such as video-based format, narrative with realistic problems (rather than a video lecture), a generative format, embedded data design, problem complexity, pairs of related adventures and links across the curriculum. The video materials serve as “anchors” (macrocontexts) for all subsequent learning and instruction. As explained by CTGV (1993), “The design of these anchors was quite different from the design of videos that were typically used in education…our goal was to create interesting, realistic contexts that encouraged the active construction of knowledge of learners. Our anchors were stories rather than lectures, and designed to be explored by students and teachers”. The video program is used as an “anchor” or situation for creating a realistic context to make learning motivating, meaningful and useful. The Taiwanese version is entitled “Mathematics in Life Series (I): Encore’s Vacation”. The program was funded by the National Science Council and is similar to The Jasper Series. With video-based format anchored instruction, the students watch the story and experience the situation more vividly, interacting with the embedded data more easily. Students have full control of their speed in watching the story as well as searching for embedded data.
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Potential for Mathematical Learning Why video? Video-based instruction enjoys several particular features that make it a useful tool for the teaching and learning of mathematics. Some of these features are explained in greater details below: A point to note here is that for the purpose of this research, video-based instruction refers to more than video. It refers to the use of video-centered (but not limited to) instruction that integrates other media elements. ? Visualization The main difficulty in the communication of mathematical concepts is due to one of the basic features of Mathematics: abstraction (Morales, 2001). Abstraction is an essential feature of Mathematics as it allows generalization of ideas and results. However, abstraction has also made Mathematics a difficult subject to teach and study. This is because most students who are trying to understand a mathematical concept lose their motivation easily when they cannot see, touch or try what they are studying. One problem they face when confronted with abstract ideas, is that they need to make the inverse way from abstraction to concrete concepts (Morales, 2001). Hence, concretization of ideas is very important in the process of learning mathematics. Visualization is one important way of concretization of mathematical ideas. Visualization is the representation of ideas, concepts or problems by images, and has always played an important role in both teaching and learning of mathematics. One of the best ways to show visualization is through video as visualization is even more effective when the images are in motion. Video here refers not only to moving images in the traditional sense, but can also include the use of computer animation, graphics, supported or integrated with software.
Figure 1 Use of computer animation to aid visualization (Project Mathematics!)
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?
Self-Paced & Exploratory Video-based instruction provides an environment where users can navigate, explore and reflect on the lessons encapsulated at their own pace. This allows learning to be individualized, according to each learner’s aptitude and understanding. To support self-paced learning environment, the content of the topic needs to be broken down into smaller, self-contained ones. For example, in the Portuguese version of “The Story of Pi” developed by Teresa Chambel from the University of Lisbon, the workbook used in the original module was converted into hypertext while preserving the underlying hierarchical structure, i.e. the structure of the sub-topics within each concept.
Figure 2. a) Text-centered page, links to video illustrations and link to video-centered page. b) Video-centered page with text index. c), d) Video-centered page with image index.
As shown in Figure 2, learners read explanation of the concept and can click on the video to see the corresponding animation. Clicking on it will focus the video, where the learner will be shown either an image-centered indexed page or a text-centered indexed page (Figure 2 a, b). These indexes make the structure and content explicit and enable the learner to proceed with his learning at his own pace. In addition, there are exercise indexes that relate the video with additional activities to explore and help understand the concepts being presented in the appropriate time and context.
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Similar self-paced, exploratory features are found in other case studies as well. In Touching Soap Films (Fig 3), the helicoid presented in the Hall of History has a link to a video description of this minimal surface. The user can navigate back to the Hall or watch the video onwards.
Figure 3 Spatio-temporal link from video to video a) Helicoid surface at the Hall of History. b) Helicoid explanation.
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Interactivity Video by itself is not sufficient for effective learning. Video images (TV, DVDs) are seen mostly in a passive way. Hence, the simple visualization of a video is not enough for an effective transmission of a concept. Learners need to interact with the instructional modules after watching the video. This interaction can be in the form of reflection, answering questions or problem-solving. That is where the use of other media and interactive elements come in. These elements not only integrate with the video, but more importantly, support and enhance its content.
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Problem-Solving Problem-solving has an essential component for mathematics learning. Video-based instruction is a suitable medium for providing and incorporating problem-solving opportunities. For example, in “The Jasper Series” developed by the Cognition and Technology Group at Vanderbilt (CTGV) and “Encore’s Vacation” developed by the Educational Technology at Tamkang University, Taiwan, they used an important design feature call “embedded data design”. Each story finish with one or more complex problems, and students were challenged to solve these problems by finding and using the relevant mathematical information presented and embedded in the video.
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Design Issues In the previous section, video-based instruction was extolled based on its particular features and affordances for visualization, self-pacing for learners, interactivity and problem-solving. Based on the literature review of video-centered instruction for mathematics learning, and the experiences of those who have designed, implemented and evaluated such programs, an effective videocentered instruction depends more than just technology. The most important element is that design of any video-centered instruction must be guided by sound learning theories. Several theories were highlighted for their relevance. These theories do not capture all the thought, nor are they completely independent. ? Situated Learning The theory of situated learning was highlighted as a guiding principle in the design of instruction in the case studies. Their emphasis on the situative nature of mathematical knowledge echoed with the readings we did earlier – that knowledge is situated, being in part a product of the activity, context ad culture in which it is developed and used (Brown, Collins & Duguid, 1989). Situated learning is based on the notion that knowledge is contextually situated and is fundamentally influenced by the given activity, context and culture (Shyu, 2000). Situated learning offers an approach to structuring learning experiences afforded by videocentered instruction in both experiential and reflective dimensions of cognition. These two cognitive dimensions – what they mean and how they were incorporated into the design of the programs – are explained in greater details in the subsequent sections. With the use of video-centered instruction and the general widespread application of technology, the ideas of situated learning can be better realized. Learning environments can now be designed with the help of technologies to provide a learning experience in school in which students can develop the skills and knowledge necessary to solve problems. Video and multimedia technology makes it possible to provide life-like inquiry situations and problem scenarios to elicit students’ problem-solving goals, strategies for solving these problems, and the connection of knowledge with daily life (Shyu, 2000). One good example is the “Encore’s Vacation” video-based program developed by the Educational Technology at Tamkang University in Taiwan. The content is designed to be consistent with the curriculum standards suggested by Taiwan’s Ministry of Education for the subject matter on Mathematics for fifth-grade students. However, it differs from the usual textbook content by anchoring the story on the adventures of a college student named Encore and three of his classmates. One example of their adventures looks like this: on a train journey back home, they encountered an accident which changed their schedule. How can they re-plan their route to get back home on time? Do they have to reschedule their trip? Why? How will they share the equal expenses? The video starts as a highly motivating third-person experience, i.e. a linear story told on video. However, as it progresses, it becomes a personal (first person) experience when students actively engage in helping the actors in the story solve a meaningful, real-life challenge. 8

In other words, the program created interesting, realistic contexts that encouraged the active construction of knowledge by learners. The use of authentic events – exposing students to the use of a mathematical domain’s conceptual tools in authentic events, resonated with our earlier readings on the importance of authentic activity. ? Scaffolding & Zone of Proximal Development Vygotsky proposed the concept “zone of proximal development” to designate the distance between the actual developmental level where problems can be solved independently and the potential level at which problems may be solve with assistance. Within this zone, the learner has an opportunity to apply skills that are in the process of developing, even though they are not yet sufficiently developed to be applied independently. Video-based instruction needs to take into account this concept of scaffolding. A welldesigned instruction should provide a framework to support the learner in the solution of a problem. One example is the interactive videodisc instructional modules developed by Ronald Henderson and Edward Landesman to teach precalculus to high school kids. The students may skip segments they see no need for, or review a concept, idea or demonstration just presented. If students have difficulty with problems and decide they need to review knowledge presented in an earlier topic within the module, they may easily review the prerequisite information. In other words, the system is designed to encourage independent solution of problems as much as possible, but at the same time, to be sensitive to the students’ needs for support and assistance (Henderson, 1993). ? Experiential Cognition There is a considerable body of older theoretical writings that have dealt extensively with the topic of having an experience (Dewey 1925, 1934/1958, James 1912/1976, 1968) in learning. Central in the theory of experiential cognition is stressing the role of the learner who constructs, sense-makes and organizes his/her knowledge by having an experience in that relevant domain of knowledge. The experiential mode leads to a state in which we perceive and react to events around us. The mind of the learner is no longer seen as passively reflective of the outside world, but as an active constructor of its own reality. Experiential thought is essential to skilled performance: it comes rapidly and effortlessly (Chambel, 2001). It marks the transition of “static” mathematical knowledge to a more “dynamic” and “processual” state. What this means is that for any effective video-based instruction, learners must not only “watch others” in experiential mode but must themselves be involved in the experience.
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However, experiencing it is not enough, for action can sometimes be confused for thoughts. To internalize ideas and new concepts, we need reflection. ? Reflective Cognition While the experiential mode of cognition can be practiced simply by experiencing it, reflection is more difficult. According to Chambel (Chambel, 2001), the reflective mode is that of comparison and contrast, of thought, of decision making. This is the mode that leads to new ideas and novel responses. Video can transmit information – ideas, concepts, problems. But it is not a neutral carrier. It affects the way we interpret and use the message, and the impact it has on us (Normal, 93). As with any technological medium, video has affordances and properties that make it easier to do some things than others. But to internalize the message, video must afford the time for reflection. More importantly, it must incorporate not only the time, but the necessary structure and organization for effective reflection. For example, if the video is designed simply to be watched in a passive mode, it will not augment reflection. However, if the learner can select what is to be seen and control the pace of the material, and it is easy to go back and forth, to stop, to make annotations, to compare and relate with other materials etc, video-based instruction can be a powerful tool for reflection. An example is the redesigned Portuguese hypervideo version of “The Story of Pi”. It incorporates effective reflection by designing flexible mechanisms to navigate and interact with the rich multimedia information spaces. It is easy to search for information and to capture the videos’ messages, through the different maps available. Interactive exercises and communication with classmates and teachers may also be woven in these hypervideo (essentially video-based hypertext) documents.
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Effects on Students’ Learning The following section summarizes from the literature review and case studies, important findings of the effects video-centered instruction has on students’ learning of mathematics. Two aspects are highlighted – students’ performance and their attitude and motivation toward mathematics. As not all the articles reviewed involved or discussed the results of implementation in detail, three relevant and sufficiently detailed papers were selected and used here.
Case Study Interactive Videodisc System (Henderson & Landesman) Computer-Video Instruction (Henderson, Landesman Kachuck) Subjects Two precalculus/trigonometry classes in high school (36 each) Topic Preparing for Calculus Time Frame 8 sub-topics, each took two to four class periods
1. Selected participants from five classes in high school (58/43 – experiment/control) & 2. Students from summer program for basic math skills competency (11/8 – factors & primes/all modules) 1. 74 fifth-graders from two classes at public elementary school in Taipei, Taiwan (for affective domain) 2. 37 fifth-graders from one class at elementary school in suburban Taipei, Taiwan (for cognitive domain) *
Fractions, Primes
Factors
and
1. Three months
2.Not stated
“Encore’s (Taiwan)
Vacation”
Problem-Solving (time, money)
1. 8 class periods in one week
2. 8 class periods in one week
* - Divided into three groups of high, medium and low ability based on their math and science score Table 1 Summary of three case studies of results on students’ learning.
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?
Performance
Interactive Videodisc System In terms of students’ performance, students who learned under a mathematics teacher and students who learned solely through the interactive videodisc materials produced comparable results in the post-tests. Evaluation responses of students who received videodisc instruction showed clearly that they felt the provision of real-world examples of mathematical concepts helped them to understand those ideas, as did the graphic representations. One interpretation of these responses is that interactive video made it possible to contextualize mathematical concepts in ways that would be very difficult to achieve without the technology. Computer-Video Instruction For the Computer-Video Instruction case study, for the first group of subjects, results indicate that students who took the computer-video modules made significantly larger improvements based on their pre and post test scores, as compared to students who did not take the modules. For the second group of subjects, which comprise of students who failed their basic skills competency tests, post test scores showed a significant improvement as compared to their pre test scores. These results show clearly that computer-video instructional modules were effective in teaching or reteaching mathematical skills and concepts to high school students who had not made normal progress in mathematical learning. “Encore’s Vacation” Only the second group of subjects is relevant here as the first group is assessed for affective domain (attitude change). Students worked in groups for problem-solving but were given the pretest and posttest individually. To assess their performance, a four-step problem-solving procedure was tested and scored in both tests. The steps were: problem-identification, problem-formulation, subgoal-generation and solution execution. This four-step problem-solving procedure was based on Polya’s mathematical problem solving model (1957). Results show that students’ mean scores of posttests improved significantly compared to pretests. All three groups (high, medium and low-ability) made improvements and there was no significant difference among the ability-groups in their increment of problem-solving skills.
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These results suggested that the instructional modules provided a more motivating environment that enhanced students’ problem-solving skills. It also indicated that all the students benefited, regardless of their mathematical and science abilities. ? Attitude & Motivation
Interactive Videodisc System To measure attitude and motivation change, an instrument was administered to assess the students’ self-perceptions of goal orientations (GO), scholastic competence (SC), and global self-worth (GWS). The GO subscale was developed to assess student goal orientations. This is based on Dweck’s model of academic motivation, which states that students with learning goal orientations are more inclined toward increasing their own competence, gain understanding or mastering something new, i.e. increased motivation. In order to examine the relation of motivational and affective variables to the learning outcomes, posttest performance for each module was regressed on these self-perception variables of GSW, SC and GO. The authors were specifically, interested in the hypothesis, derived from Dweck’s research, that students with learning goal orientations would experience more favorable learning outcomes than would students with performance goal orientations. Results indicated that Goal Orientation is the strongest predictor of learning outcomes, as measured by posttests. This suggests that student goal orientations may play an important role in academic learning. Computer-Video Instruction In comparison, there were no similar instruments used in this case study. However, qualitative studies done showed that students who learned from the computer-video modules believed they had learned from the experience and they felt very positive about it. In addition, there was some support from the field trial results and responses that exposure to the materials would help students recognize that it is possible for them to learn mathematics, and that this was reflected by positive changes in their effort attributions specific to mathematics. In other words, it means the students felt that if they are willing to put in more effort, they can see a difference in their mathematics performance.
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“Encore’s Vacation” Results from the field tests for the first group of subjects showed a significant effect on student attitudes toward instruction. Analyses showed that students, after receiving the new instruction, felt more positive about, interested in and less anxious toward mathematics. In terms of the difference in effect between genders, results indicated there were no significant differences between girls and boys in their attitudes toward mathematics.
Conclusion This research sets out to study some of the case studies of the use of video-based instruction for the teaching and learning of mathematics, and the issues involved in it. From the results gathered, video has shown to have great potential. Integrating with other multimedia elements and computer animation, it helps to concretize concepts by visualization. It can be interactive, allows student to control their own pace of learning, and is a suitable medium for providing problemsolving opportunities. However, to harness the power video-based instruction provides, certain design principles were highlighted in the case studies. Situated learning and providing zonal of proximal support are important theoretical guideposts. For effective learning, learners must reflect as well as experience. Video course material, as rich as it is, is better used if time and structure are provided for learners’ reflection. One important consideration when designing and implementing such instructional modules is that learning processes have to face heterogeneous learners, with diverse learning styles and approaches. Results have shown that students of video-based instruction learned at least as well as students attending traditional classrooms learning. This is reflected in the performances in some of the case studies. Students’ attitudes and motivation were reflected in a positive way as well. To conclude, how we integrate interaction processes, authoring approaches and rich elements of multimedia information to create an effective learner-centered educational environment for the learning of mathematics calls for more than just technology. Cognition and learning theories play an important, fundamental role in its design. The research suggests that the adoption of videobased instruction in our pedagogical framework is feasible and potentially effective; that videobased instruction can not only be a valuable pedagogical tool that reveals mathematics for what it is, not only understandable and exciting, but eminently worthwhile as well.
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Recommendations for Further Research The case studies highlighted here involve the use of video-based instruction delivered through videotapes and DVDs. For the “Project Mathematics!” video modules, although its Portuguese version integrated with other multimedia elements and hyperlinks, the videos were not assessed via the internet. Hence, one area I would suggest, and forsee, for possible future research is the use of the internet as a delivery medium for mathematics learning. The dynamic and interactive nature of the web has tremendous implication in terms of design and instruction effectiveness. For example, lessons can be designed in such a way that students can give feedback or post questions for particular segments of the video on the topic. Based on these feedback or the assessments, teachers can fine-tune or make changes to the video to improve on its effectiveness. This is but one potential and difference between the affordances of online video versus traditional delivery mode; and is definitely an area worth further research. The potential impact of the use of video-centered instruction on teachers’ pedagogy, curriculum design and students’ learning – three essential components of math education – needs further investigation and study as well. A supporting framework will help in the integration of such instructions and bring about effective learning experiences for the students.
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References Apostol T., Computer Animated Mathematics Videotapes, J. Borwein, M. Morales, et.al. (Eds.), Multimedia Tools for Communicating Mathematics, Springer Verlag, 2001 Apostol T., Using Computer Animation to Teach Mathematics, CBMS Issues in Mathematics Education Volume 3, 1993 13-38 Browm, J., Collins, A., and Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1). 32-42 Chambel T. and Guimaraes N., Communicating and Learning Mathematics with Hypervideo, J. Borwein, M. Morales, et.al. (Eds.), Multimedia Tools for Communicating Mathematics, Springer Verlag, 2001 Chambel T. and Guimaraes N., Context Perception in Video-Based Hypermedia Spaces, HT’ 02, June 11-15, 2002, 85-93 Henderson R. and Landesman E., The Interactive Videodisc System in the Zone of Proximal Development: Academic Motivation and Learning Outcomes in Precalculus, J. Educational Computing Research, Vol. 9(1) 29-43, 1993 Henderson R., Landesman E. and Kachuck I., Computer-Video Instruction in Mathematics: Field Test of an Interactive Approach, Journal for Research in Mathematics Education Vol. 16, No. 3 207-224, 1985 Morales M., Hypervideo as a tool for communicating Mathematics Normal, D., “Things That Make Us Smart”, Addison Wesley Publishing Company, 1993 Shyu H., Using video-based anchored instruction to enhance learning: Taiwan’s experience, British Journal of Educational Technology, Vol 31 No. 1 57-69, 2000
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A.63 8 B.64 C.328 D.24 3.在Rt △ABC 中，∠C=900，AB=2AC ，在BC 上取一点D ，使AC=CD ，则CD ：BD=（ ） A.213+ B.13- C.2 3 D.不能确定 4.在Rt △ABC 中，∠C=900，∠A=300，b=310，则a= ，c= ； 5.已知在直角梯形ABCD 中，上底CD=4，下底AB=10，非直角腰BC=34， 则底角∠B= ； 6.若∠A 是锐角，且cosA=5 3，则cos （900-A ）= ； 7.在Rt △ABC 中，∠C=900，AC=1，sinA= 23，求tanA ，BC ． 8.在△ABC 中，AD ⊥BC ，垂足为D ，AB=22，AC=BC=52，求AD 的长． 9. 去年某省将地处A 、B 两地的两所大学合并成一所综合性大学，为了方便两地师生交往，学校准备在相距2km 的A 、B 两地之间修一条笔直的公路，经测量在A 地北偏东600方向，B 地北偏西450方向的C 处有一个半径为0.7km 的公园，问计划修筑的这条公路会不会穿过公园？为什么？ 第28课时 锐角三角函数的简单应用 【知识梳理】 1. 坡面与水平面的夹角(α)称为坡角,坡面的铅直高度与水平宽度的比称为坡度i(或坡比),即坡度等于坡角的正切值． 2. 仰角：仰视时，视线与水平线的夹角． 俯角：俯视时，视线与水平线的夹角． 【思想方法】 1. 常用解题方法——设k 法 2. 常用基本图形——双直角 A B C D C A B 第8题图 第9题图

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； 6．（10分）一个圆柱形容器的容积为V立方米，开始用一根小水管向容器内注水，水面高度达到容器高度一半后，改用一根口径为小水管2倍的大水管注水．向容器中注满水的全过程共用时间t分．求两根水管各自注水的速度． 7．（10分）（2009?郴州）如图1，已知正比例函数和反比例函数的图象都经过点M（﹣2，﹣1），且P（﹣1，﹣2）为双曲线上的一点，Q为坐标平面上一动点，PA垂直于x轴，QB垂直于y轴，垂足分别是A、B． （1）写出正比例函数和反比例函数的关系式； （2）当点Q在直线MO上运动时，直线MO上是否存在这样的点Q，使得△OBQ与△OAP面积相等如果存在，请求出点的坐标，如果不存在，请说明理由； （3）如图2，当点Q在第一象限中的双曲线上运动时，作以OP、OQ为邻边的平行四边形OPCQ，求平行四边形 OPCQ周长的最小值． 8．（10分）（2008?海南）如图，P是边长为1的正方形ABCD对角线AC上一动点（P与A、C不重合），点E在线段BC上，且PE=PB． （1）求证：①PE=PD；②PE⊥PD； （2）设AP=x，△PBE的面积为y． 、 ①求出y关于x的函数关系式，并写出x的取值范围； ②当x取何值时，y取得最大值，并求出这个最大值．

《从爱因斯坦到霍金的宇宙》考试答案

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《从爱因斯坦到霍金的宇宙》期末考试（20）
一、 单选题（题数：50，共 1 哈勃常数 50.0 分）
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A、随时间的增加而变大 B、随时间的增加而减小 C、随距离的增加而增大 D、随距离的增加而减小 我的答案：B 2 以下哪项说法正确
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A、牛顿和惠更斯都支持光的微粒说 B、牛顿支持光的微粒说，惠更斯支持光的波动说 C、牛顿支持光的波动说，惠更斯支持光的微粒说 D、牛顿和惠更斯都支持光的波动说 我的答案：B 3 时序保护思想是谁提出的
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A、爱因斯坦 B、霍金 C、彭罗斯 D、朗道 我的答案：B 4 约翰·米歇尔在（）年提出了暗黑的概念。（）
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A、1783 B、1784 C、1785 D、1786 我的答案：A 5 最早记载牛顿和苹果故事的是
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A、伏尔泰 B、卢梭 C、霍布斯 D、胡克 我的答案：A 6 爱因斯坦方程又叫
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A、场方程 B、惯性方程

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C、非欧方程 D、引力方程 我的答案：A 7 以下哪位科学家没有和杨振宁共同做出过重大的科学成就
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A、米尔斯 B、巴克斯特 C、邓稼先 D、李政道 我的答案：C 8 肉眼可以看见的除太阳之外的最亮的恒星是哪一颗？（）
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A、牛郎星 B、织女星 C、天狼星 D、北极星 我的答案：C 9 第一次完成环球航行的航海家是谁？（） 1.0 分
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A、马可波罗 B、郑和 C、哥伦布 D、麦哲伦 我的答案：D 10 最早的超新星爆发记录是在中国的哪个朝代？（）
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A、商代 B、唐代 C、宋代 D、明代 我的答案：C 11 爱因斯坦提出相对论主要参考了哪个实验
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A、迈克尔逊实验 B、斐索实验 C、洛伦兹实验 D、庞加莱实验 我的答案：B 12 物理学家拉普拉斯是哪个国家的人？（）
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A、德国

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2）解直角三角形的基本类型 注：有斜用弦，无斜用切，宁乘勿除，取原避中，化斜为直． （3）几种常见的三角形： 考点四：解直角三角形的应用 1．相关概念： （1 ）仰角和俯角：它们都是视线与水平线所成的角，如图4—2—83（a）所示，视线在水平线上方的 角叫作仰角，视 线在水平线下方的角叫作俯角． （2）坡度与坡角：如图4—2—83（b）所示，坡面的垂直高度 h和水平宽度 l 的比叫作坡度（坡 比）．用字母 i表示，即i h．把坡面与水平面的夹角，记作（叫作坡角），那么i h＝tan ．ll （3 ）指北或指南方向线与与目标方向线所成的小于90°的角，叫作方向角．如图4—2—83（c）所示，OA，OB，OC， OD 的方向角分别为：北偏东30 °，南偏东45 °（东南方向），南偏西30°，北偏西45°（西北方向）．

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2018从爱因斯坦到霍金的宇宙.超星尔雅答案.最新

物理学的开端：经验物理时期已完成成绩：分 【单选题】相对论分为狭义相对论和广义相对论，它们都是关于（）的基本理论。 A、引力和重力 B、时空和重力 C、时间和空间 D、时空和引力 我的答案：D得分：分 2 【单选题】提出“格物穷理”的是谁（） A、张载 B、陆九渊 C、朱熹 D、王阳明 我的答案：C得分：分 3 【判断题】 中国奴隶社会比欧洲时间短，西方封建社会比中国时间短。（） 我的答案：√得分：分 4 【判断题】欧几里得的学生是阿基米德的老师。（） 我的答案：√得分：分 伽利略与经典物理的诞生已完成成绩：分 1 【单选题】惯性定律认为物体在受任何外力的作用下，不会保持下列哪种运动状态（） A、匀速曲线 B、匀速直线 C、加速直线 D、加速曲线

我的答案：B得分：分 2 【单选题】伽利略有许多成就，但不包括下面哪一项（） A、重述惯性定律 B、阐述相对性原理 C、发现万有引力 D、自由落体定律 我的答案：C得分：分 3 【单选题】认为万物都是由原子构成的古希腊哲学家是谁() A、德谟克利特 B、毕达哥拉斯 C、色诺芬 D、亚里士多德 我的答案：A得分：分 4 【判断题】奥地利物理学家伽利略是近代实验科学的先驱者。（） 我的答案：×得分：分 经典物理的三大支柱：经典力学、经典电动力学、经典热力学和统计力学已完成成绩：分 1 【单选题】继发现热力学第一定律和第二定律后，有谁发现了“热力学第三定律”。（） A、克劳修斯 B、开尔文 C、能斯特 D、焦耳 我的答案：C得分：分 2 【多选题】下列选项不属于经典物理学范畴的是（）。

A、万有引力定律 B、热质学说 C、量子论 D、狭义相对性原理 我的答案：B、C、D得分：25分 3 【判断题】根据双缝干涉实验，牛顿提出了光学上的“波动说”。（） 我的答案：×得分：分 4 【判断题】根据“热力学第零定律”，两个热力学系统彼此处于热平衡的前提条件是每一个都与第三个热力学系统处理热平衡。（） 我的答案：√得分：分 经典物理的局限与量子论的诞生已完成成绩：分 1 【单选题】物理学上用紫外灾难形容经典理论的困境，其具体内容指()。 A、维恩线在短波波段与实验值的巨大差异 B、瑞利-金斯线在短波波段与实验值的巨大差异 C、维恩线在长波波段与实验值的巨大差异 D、瑞利-金斯线在长波波段与实验值的巨大差异 我的答案：B得分：分 2 【单选题】与平衡热辐射实验值在长波和短波波段都吻合是哪条线() A、普朗克线 B、维恩线 C、瑞丽-金斯线 D、爱因斯坦线 我的答案：A得分：分 3

中考数学-特殊角三角函数的应用

中考数学 特殊角三角函数的应用 1师生共同完成课本第82页例3:求下列各式的值. (1)COS260°sin260 ° COS45 o (2)-tan45 ° . sin 45 教师以提问方式一步一步解上面两题?学生回答，教师板书. 1 ?3 解：(1} COS260 °sin260° =(2)2+(乙)2=1 ⑵ CO^-ta n45 ° =上 + 2-1=0 sin 45 2 2 2?师生共同完成课本第82页例4:教师解答题意: (1)如课本图28? 1-9 ( 1),在Rt△ ABC 中，/ C=90, AB= J6 , BC= J3，求/ A 的度数. (2)如课本图28. 1-9 (2),已知圆锥的高AO等于圆锥的底面半径OB的J3倍，求 a. 教师分析解题方法：要求一个直角三角形中一个锐角的度数，可以先求它的某一个三角函数的值，如果这个值是一个特殊解，那么我们就可以求出这个角的度数. 解：(1)在课本图28. 1-9 (1)中, BC 73 血 -sinA= —= AB V6 2

(2)在课本图28 . 1-9 (2)中， AO y/30B庁 ■/ tana= =、、3 , OB OB ??? a=60°. 教师提醒学生：当A、B为锐角时，若A丰B,则si nA 丰 si nB , cosA 丰 cosB, tanA 丰 ta nB.随堂练习 学生做课本第83页练习第1、2题. 课时总结 学生要牢记下表: 对于sina与tana, . 教后反思 第3课时作业设计

课本练习

做课本第85页习题28. 1复习巩固第3题. 双基与中考 （本练习除了作为本课时的课外作业之外，余下的部分作为下一课时（习题课）学生 的课堂作业?学生可以自己根据具体情况划分课内、课外作业的份量） 、选择题. 1 .已知: Rt △ ABC 中，/ 3 C=90 , cosA=—, 5 AB=15 , 则AC 的长是（） A . 3 B . 6 C . 9 D . 12 2.下列各式中不正确的是（ ）. A . si n 260 °+COS 260° =1 B . si n30° +cos30° =1 C . sin35 ° =cos55 ° D . tan 45° >sin45 ° 3 .计算 2sin30 ° -2cos60° +ta n45 °的结果是（ ）. A . 2 B . 3 C . 、、2 D . 1 1 cosA w ，那么（ ） 2 B . 60°60°时，cosa 的值（）. 4. 已知/ A 为锐角，且 C . 0°

中考数学—锐角三角函数的综合压轴题专题复习及答案

一、锐角三角函数真题与模拟题分类汇编（难题易错题） 1．已知：如图，在四边形 ABCD 中， AB ∥CD ， ∠ACB =90°， AB=10cm ， BC=8cm ， OD 垂直平分 A C ．点 P 从点 B 出发，沿 BA 方向匀速运动，速度为 1cm/s ；同时，点 Q 从点 D 出发，沿 DC 方向匀速运动，速度为 1cm/s ；当一个点停止运动，另一个点也停止运动．过点 P 作 PE ⊥AB ，交 BC 于点 E ，过点 Q 作 QF ∥AC ，分别交 AD ， OD 于点 F ， G ．连接 OP ，EG ．设运动时间为 t ( s )（0＜t ＜5） ，解答下列问题： （1）当 t 为何值时，点 E 在 BAC 的平分线上？ （2）设四边形 PEGO 的面积为 S(cm 2) ，求 S 与 t 的函数关系式； （3）在运动过程中，是否存在某一时刻 t ，使四边形 PEGO 的面积最大？若存在，求出t 的值；若不存在，请说明理由； （4）连接 OE ， OQ ，在运动过程中，是否存在某一时刻 t ，使 OE ⊥OQ ？若存在，求出t 的值；若不存在，请说明理由． 【答案】（1）4s t =；（2）PEGO S 四边形2 31568 8 t t =-+ + ，(05)t <<；（3）5 2t =时， PEGO S 四边形取得最大值；（4）16 5 t = 时，OE OQ ⊥. 【解析】 【分析】 （1）当点E 在∠BAC 的平分线上时，因为EP ⊥AB ，EC ⊥AC ，可得PE=EC ，由此构建方程即可解决问题． （2）根据S 四边形OPEG =S △OEG +S △OPE =S △OEG +（S △OPC +S △PCE -S △OEC ）构建函数关系式即可． （3）利用二次函数的性质解决问题即可． （4）证明∠EOC=∠QOG ，可得tan ∠EOC=tan ∠QOG ，推出EC GQ OC OG =，由此构建方程即可解决问题． 【详解】 （1）在Rt △ABC 中，∵∠ACB=90°，AB=10cm ，BC=8cm ， ∴22108-=6（cm ）， ∵OD 垂直平分线段AC ， ∴OC=OA=3（cm ），∠DOC=90°， ∵CD ∥AB ，

初二数学经典难题及答案

A P C D B 初二数学经典题型 1．已知：如图，P 是正方形ABCD 内点，∠PAD ＝∠PDA ＝150 ．求证：△PBC 是正三角形． 证明如下。 首先，PA=PD ，∠PAD=∠PDA=（180°-150°）÷2=15°，∠PAB=90°-15°=75°。 在正方形ABCD 之外以AD 为底边作正三角形ADQ ， 连接PQ ， 则 ∠PDQ=60°+15°=75°，同样∠PAQ=75°，又AQ=DQ,，PA=PD ，所以△PAQ ≌△PDQ ， 那么∠PQA=∠PQD=60°÷2=30°，在△PQA 中， ∠APQ=180°-30°-75°=75°=∠PAQ=∠PAB ，于是PQ=AQ=AB ， 显然△PAQ ≌△PAB ，得∠PBA=∠PQA=30°， PB=PQ=AB=BC ，∠PBC=90°-30°=60°，所以△ABC 是正三角形。 2.已知：如图，在四边形ABCD 中，AD ＝BC ，M 、N 分别是AB 、CD 的中点，AD 、BC 的延长线 交MN 于E 、F ．求证：∠DEN ＝∠F ． 证明:连接AC,并取AC 的中点G,连接GF,GM. 又点N 为CD 的中点,则GN=AD/2;GN ∥AD,∠GNM=∠DEM;(1) 同理:GM=BC/2;GM ∥BC,∠GMN=∠CFN;(2) 又AD=BC,则:GN=GM,∠GNM=∠GMN.故:∠DEM=∠CFN. 3、如图，分别以△ABC 的AC 和BC 为一边，在△ABC 的外侧作正方形ACDE 和正方形CBFG ，点P 是EF 的中点．求证：点P 到边AB 的距离等于AB 的一半． 证明：分别过E 、C 、F 作直线AB 的垂线，垂足分别为M 、O 、N ， 在梯形MEFN 中，WE 平行NF 因为P 为EF 中点，PQ 平行于两底 所以PQ 为梯形MEFN 中位线， 所以PQ ＝（ME ＋NF ）/2 又因为，角0CB ＋角OBC ＝90°＝角NBF ＋角CBO 所以角OCB=角NBF 而角C0B ＝角Rt ＝角BNF CB=BF 所以△OCB 全等于△NBF △MEA 全等于△OAC （同理） 所以EM ＝AO ，0B ＝NF 所以PQ=AB/2. 4、设P 是平行四边形ABCD 内部的一点，且∠PBA ＝∠PDA ．求证：∠PAB ＝∠PCB ． 过点P 作DA 的平行线，过点A 作DP 的平行线，两者相交于点E ；连接 BE

爱因斯坦到霍金的宇宙期末考试题及答案

从爱因斯坦到霍金的宇宙期末考试题及答案 1“吾爱吾师，吾更爱真理”这句话是谁说的？（） A、苏格拉底 B、柏拉图 C、亚里士多德 D、色诺芬 正确答案： C 2 下列人物中最早使用“物理学”这个词的是谁？（） A、牛顿 B、伽利略 C、爱因斯坦 D、亚里斯多德 正确答案： D 3 “给我一个支点，我就可以耗动地球”这句话是谁说的？（） A、欧几里得 B、阿基米德 C、亚里士多德 D、伽利略 正确答案： B

4 “格物穷理”是由谁提出来的？（） A、张载 B、朱熹 C、陆九渊 D、王阳明 正确答案： B 5 相对论是关于（）的基本理论，分为狭义相对论和广义相对论。 A、时空和引力 B、时空和重力 C、时间和空间 D、引力和重力 正确答案： A 欧洲奴隶社会比中国时间长，中国封建社会比西方时间长。 正确答案：√ 7 阿基米德是欧几里得的学生的学生。 正确答案：√ 8 西方在中世纪有很多创造。 正确答案：×

伽利略与经典物理的诞生 1 以下不属于伽利略的成就的是（） A、重述惯性定律 B、发现万有引力 C、阐述相对性原理 D、自由落体定律 正确答案： B 2 “地恒动而人不知，譬如闭舟而行，不觉舟之运也”体现了什么物理学原理？（） A、相对性原理 B、惯性原理 C、浮力定理 D、杠杆原理 正确答案： A 3 哪位古希腊哲学家认为万物都是由原子构成的 A、亚里士多德 B、毕达哥拉斯 C、色诺芬 D、德谟克利特

正确答案： D 4 惯性定律认为物体在不受任何外力的作用下，会保持下列哪种运动状态？（） A、匀速曲线 B、加速直线 C、匀速直线 D、加速曲线 正确答案： C 5 《关于托勒密和哥白尼两大世界体系的对话》与《天体运行论》都是伽利略的著作正确答案：× 伽利略是奥地利物理学家，近代实验科学的先驱者。（） 正确答案：× 7 伽利略的逝世和牛顿的出生都是在1642年。（） 正确答案：√ 8 伽利略认为斜面上的运动是冲淡了的自由落体运动。（） 正确答案：√ 经典物理的三大支柱：经典力学、经典电动力学、经典热力学和统计力学

中考数学三角函数练习题

和 三角函数专项训练 1．（2009眉山）海船以5海里/小时的速度向正东方向行驶，在A处看见灯塔B在海船的北偏东60°方向，2小时后船行驶到C处，发现此时灯塔B在海船的北偏西45方向，求此时灯塔B到C处的距离． 2．（2009年中山）如图所示，A．B两城市相距100km，现计划在这两座城市间修建一条高速公路（即线段AB），经测量，森林保护中心P在A城市的北偏东30°B城市的北偏西45°的方向上，已知森林保护区的范围在以P点为圆心，50km为半径的圆形区域内，请问计划修建的这条高速公路会不会穿越保护区，为什么？（参考数据： 3≈1.732，2≈1.414） 3．（2009年哈尔滨）如图，一艘轮船以每小时20海里的速度沿正北方向航行，在A处测得灯塔C在北偏西30°方向，轮船航行2小时后到达B处，在B处测得灯塔C在北偏西60°方向．当轮船到达灯塔C的正东方向的D处时，求此时轮船与灯塔C的距离．（结果保留根号） 北 C D 60° B 30° A 4．（2009年凉山州）如图，要在木里县某林场东西方向的两地之间修一条公路MN，已知C点周围200米范围内为原始森林保护区，在MN上的点A处测得C在A的北偏东45°方向上，从A向东走600米到达B处，测得C在点B的北偏西60°方向上． （1）MN是否穿过原始森林保护区？为什么？（参考数据：3≈1.732） （2）若修路工程顺利进行，要使修路工程比原计划提前5天完成，需将原定的工作效率提高25%，则原计划完成这项工程需要多少天？

. C M A B N （第21题） 5．（2009年辽宁省锦州）为了加快城市经济发展，某市准备修建一座横跨南北的大桥如图10所示，测量队在点A处观测河对岸水边有一点C，测得C在北偏东60°的方向上，沿河岸向东前行30米到达B处，测得C在北偏东45°的方向上，请你根据以上数据帮助该测量队计算出这条河的宽度.(结果保留根号) 6．（2009年湖南长沙）某校九年级数学兴趣小组的同学开展了测量湘江宽度的活动．如图，他们在河东岸边的A点测得河西岸边的标志物B在它的正西方向，然后从A点出发沿河岸向正北方向行进550米到点C处，测得B在点C的南偏西60°方向上，他们测得的湘江宽度是多少米？（结果保留整数，参考数据：2≈1.414，3≈1.732）北 西东 南 C B A 7．（2009山西省太原市）如图，从热气球C上测得两建筑物A．B底部的俯角分别为30°和60°．如果这时气球的高度CD为90米．且点A．D．B在同一直线上，求建筑物A．B间的距离． E C F 30°60° A D B

初二数学提高题[附答案]

初二数学提高题[附答案]

综合题 1．如图（1），直角梯形OABC 中，∠A= 90°，AB ∥CO, 且AB=2，OA=2，∠BCO= 60°。 （1）求证：OBC 为等边三角形；（2）如图（2），OH ⊥BC 于点H ，动点P 从点H 出发，沿线 段HO 向点O 运动，动点Q 从点O 出发，沿线段OA 向 点A 运动，两点同时出发，速度都为1/秒。设点P 运动的时间为t 秒，ΔOPQ 的面积为S ，求S 与t 之 间的函数关系式，并求出t 的取值范围； （3）设PQ 与OB 交于点M ，当OM=PM 时，求t 的值。3?图(1)60?B C A o 图(2)60?M P Q H B A （备用图）H 60? B C A

333 33333解：1)根据勾股定理，AB=2，OA=2，则BO=4=2AB ，所以△ABO 是一个30°60°90°的三角形。 ∵AB//CO ，∠A=90°∴∠AOC=180°-90°=90° ∵∠AOB=30°，∴∠BOC=90°-30°=60°=∠C ∴△OBC 为等边三角形 2)∵点P 运动的时间为t 秒，∴OQ=PH=t ∵OH ⊥BC ，∴∠CHO=90°， ∴∠COH=30°，OH=( /2)BC=2 ∴∠QOP=60°，OP=2 -t ∴S=1/2t(2 -t)× /2=3/2t- /4t 2，且(0

2. 如图，正比例函数图像直线l经过点A（3，5），点B 在x轴的正半轴上，且∠ABO＝45°。AH⊥OB，垂足为点H。　（1）求直线l所对应的正比例函数解析式； 　（2）求线段AH和OB的长度； 　（3）如果点P是线段OB上一点，设OP＝x，△APB的面积为S，写出S与x的函数关系式，并指出自变量x 的取值范围。 解：1)设y=kx为正比例解析式，当x=3,y=5时，3k=5,k=5/3 2)AH即A的纵坐标，∴AH=5 ∵AH⊥BH，∠ABH=45°，∴∠HAB=∠ABH=45°，∴AH=BH=5 OH即A的横坐标，∴OH=3 ∵OB=OH+BH，∴OB=5+3=8 3)∵OB=8，OP=x，∴BP=8-x

中考数学三角函数应用题-(1)

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中考数学复习专题三角函数与圆

2011中考数学复习专题—三角函数和圆 考点1 三角形的边角关系 主要考查：三种锐角三角函数的概念，特殊值计算，锐角函数之间的关系，解直角三角形及应用。 1.如图所示 ，Rt △ABC ～Rt △DEF ，则cosE 的值等于（ ） A ．2 1 B ．2 2 C ．2 3 D ．33 2.如图，已知直角三角形ABC 中，斜边AB 的长为m ，∠B=ο40，则直角边BC 的长是（ ） A ．ο40sin m B ．ο40cos m C ．ο40tan m D ．ο40tan m 3.王师傅在楼顶上的点A 处测得楼前一棵树CD 的顶端C 的俯角为ο60，又知水平距离BD=10m ，楼高AB=24m ，则树高CD 为（ ） A ．()m 31024- B ．m ???? ??-331024 C ．()m 3524- D ．9m 4.如图是掌上电脑设计用电来测量某古城墙高度的示意图。点P 处放一水平的平面镜，光线从点A 出发经平面镜反射后刚好射到古城墙CD 的顶端C 处，已知AB ⊥BD ，CD ⊥BD ，且测得AB=1.2米，BP=1.8米，PD=12米，那么该古城墙的高度是（ ） A ．6米 B ．8米 C ．18米 D ．24米 5.如图所示，某河堤的横断面是梯形ABCD ，BC ∥AD ，迎水坡AB 长13米，且tan ∠BAE= 512，则河堤的高BE 为 米。 6．如果，小明同学在东西方向的环海路A 处，测得海中灯塔P 在北偏东ο60方向上，在A 处东500米的B 处，测得海中灯塔P 在北偏东ο30方向上，则灯塔P 到环海路的距离 PC= 米（用根号表示）。

中考数学——三角函数专题

三角函数1 一．解答题（共10小题） 1．如图：一辆汽车在一个十字路口遇到红灯刹车停下，汽车里的驾驶员看地面的斑马线前后两端的视角分别是∠DCA=30°和∠DCB=60°，如果斑马线的宽度是AB=3米，驾驶员与车头的距离是0.8米，这时汽车车头与斑马线的距离x是多少？ 2．一种拉杆式旅行箱的示意图如图所示，箱体长AB=50cm，拉杆最大伸长距离BC=35cm，（点A、B、C在同一条直线上），在箱体的底端装有一圆形滚轮⊙A，⊙A与水平地面切于点D，AE∥DN，某一时刻，点B距离水平面38cm，点C距离水平面59cm． （1）求圆形滚轮的半径AD的长； （2）当人的手自然下垂拉旅行箱时，人感觉较为舒服，已知某人的手自然下垂在点C处且拉杆达到最大延伸距离时，点C距离水平地面73.5cm，求此时拉杆箱与水平面AE所成角∠CAE的大小（精确到1°，参考数据：sin50°≈0.77，cos50°≈0.64，tan50°≈1.19）．

3．如图，书桌上的一种新型台历和一块主板AB、一个架板AC和环扣（不计宽度，记为点A）组成，其侧面示意图为△ABC，测得AC⊥BC，AB=5cm，AC=4cm，现为了书写记事方便，须调整台历的摆放，移动点C至C′，当∠C′=30°时，求移动的距离即CC′的长（或用计算器计算，结果取整数，其中=1.732，=4.583） 4．某课桌生产厂家研究发现，倾斜12°～24°的桌面有利于学生保持躯体自然姿势．根据这一研究，厂家决定将水平桌面做成可调节角度的桌面．新桌面的设计图如图1，AB可绕点A 旋转，在点C处安装一根可旋转的支撑臂CD，AC=30cm． （1）如图2，当∠BAC=24°时，CD⊥AB，求支撑臂CD的长； （2）如图3，当∠BAC=12°时，求AD的长．（结果保留根号） （参考数据：sin24°≈0.40，cos24°≈0.91，tan24°≈0.46，sin12°≈0.20）