• Journal for History of Mathematics
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• v.20 no.3
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• pp.147-166
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• 2007

This study is about the law of cosines. It dealt with its historical origin and the developmental process of the age of Greece, Islam and Modern age. Especially, we tried to find out how the extension of the law of cosines for spherical triangles and tetrahedron from the law of cosines for plane was done. On the basis of this analysis, we investigated how the law of cosines was generated and proved it through the logical demonstration and mathematical induction. This made us find out the mathematical meaning of mathematical concepts.

• The Pure and Applied Mathematics
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• v.4 no.1
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• pp.1-6
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• 1997

We will construct the generalized law of cosines in a tetrahedron, in a natural way, which gives three dimensional Pythagoras' theorem and enables us to calculate the volume of an arbitrary tetrahedron.

• The Pure and Applied Mathematics
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• v.16 no.4
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• pp.359-367
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• 2009

The graph of the parallelogram law is well known, which gives rise to the characterization of an inner product space among normed linear spaces [6]. In this paper we will sketch graphs of its deformations according to our previous paper [7, Theorem 3.1 and 3.2]; each one of which characterizes an inner product space among normed linear spaces. Consequently, the graphs of some classical characterizations of an inner product space follow easily.

• Journal for History of Mathematics
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• v.36 no.2
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• pp.21-34
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• 2023

Spherical trigonometry refers to the geometry related to spherical triangles. It has been an important tool for studying astronomy since ancient times. In trigonometry, concepts such as trigonometric functions naturally emerge from the relationship between arcs and chords of a circle. In this paper, we briefly examine the origin of spherical trigonometry. To introduce the basics of spherical trigonometry, we present fundamental and important theorems such as Menelaus's theorem, the law of sines and the law of cosines on a sphere, along with their proofs. Furthermore, we discuss the educational value and potential applications of spherical trigonometry.

• Communications of Mathematical Education
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• v.21 no.1 s.29
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• pp.51-64
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• 2007

In this study we generalize the law of cosine(in any triangle the square of one side is equal to the sum of the squares of the other sides minus twice their product times the cosine of their included angle), We find the following generalized law of cosine: in any polygon the square of one side is equal to the sum of the squares of the other sides minus twice their products times the cosines of their included angles, and prove it using vector.

• Journal of Educational Research in Mathematics
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• v.24 no.2
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• pp.103-124
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• 2014

The process of analogical reasoning can be conventionally summarized in five steps : Representation, Access, Mapping, Adaptation, Learning. The purpose of this study is to develop more detailed model for reason of analogies considering the distinct characteristics of the mathematical education based on the process of analogical reasoning which is already established. Ultimately, This model is designed to facilitate students to use analogical reasoning more productively. The process of developing model is divided into three steps. The frist step is to draft a hypothetical model by looking into historical example of Leonhard Euler(1707-1783), who was the great mathematician of any age and discovered mathematical knowledge through analogical reasoning. The second step is to modify and complement the model to reflect the characteristics of students' thinking response that proves and links analogically between the law of cosines and the Pythagorean theorem. The third and final step is to draw pedagogical implications from the analysis of the result of an experiment.

• Journal of Educational Research in Mathematics
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• v.15 no.2
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• pp.131-145
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• 2005

This study intends to analyze didactically on triangle-determining conditions and triangle-congruence conditions. The result of this study revealed the followings: Firstly, many pre-service mathematics teachers and secondary school students have insufficient understanding or misunderstanding on triangle-determining conditions and triangle-congruence conditions. Secondly, the term segment instead of edge may show well the concern of triangle-determining conditions. Thirdly, when students learn the method of finding six elements of triangle using the law of sines and cosines in high school, they should be given the opportunity to reflect the relation and the difference between triangle-determining situation and the situation of finding six elements of triangle. Fourthly, accepting some conditions like SSA-obtuse as a triangle-determining condition or not is not just a logical problem. It depends on the specific contexts investigating triangle-determining conditions. Fifthly, textbooks and classroom teaching need to guide students to discover triangle-deter-mining conditions in the process of inquiry from SSS, SSA, SAS, SAA, ASS, ASA, AAS, AAA to SSS, SAS, ASA, SAA. Sixthly, it is necessary to have students know the significance of 'correspondence' in congruence conditions. Finally, there are some problems of using the term 'correspondent' in describing triangle-congruence conditions.

• The Korean Journal of Applied Statistics
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• v.16 no.2
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• pp.283-291
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• 2003

The Gini index is one of the most commonly used measures of inequality of income distributions. In this paper, the Lorenz curve is estimated by arcs of two optimal circles, and a new simple method to estimate the Gini index is proposed using the law of cosines. We compare the proposed estimator with the estimator proposed by Ogwang and Rao(1996) in terms of the mean squared error(MSE) though Monte Carlo simulation in a Pareto distribution.

• Communications of Mathematical Education
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• v.35 no.2
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• pp.193-212
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• 2021

This study is to support the school's mathematics exploration activities. Mathematics exploration is a very important mathematical activity not only for mathematics teachers, but also for students. Looking at the development of mathematics, it has been extended from one mathematical fact to a new mathematical fact. Mathematics exploration activities are not unique to mathematicians, and opportunities are equally given to all ordinary people who are learning mathematics and teaching mathematics. Therefore, the purpose of this study is to develop a method of mathematics exploration activities that teachers and students can perform in schools, based on mathematics exploration activities based on one mathematical fact. Specifically, the cosine law was selected as one mathematical fact, and mathematical exploration activities were performed based on the cosine law. By analyzing the results of these mathematics exploration activities, we developed a method to explore school mathematics. Through the results of this study, it is expected that mathematics exploration activities will be conducted equally by students and teachers in the mathematics classroom.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.34 no.12B
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• pp.1387-1398
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• 2009

In the latest we will make a demand for the precision position estimation for the Mobile-station(MS)'s position. But, we have a lot of problems the position estimation method using the existing method. The Base-station(BS) measure a distance according to time delay waves to receive propagate from the MS and estimate the position using the existing circle equation with method to be selected BSs in close proximity the MS. It knows that happens a lot of error the estimated position and the true position. This paper propose that the method is selected round BSs to estimate for MS's position and estimated the angle using the second law of cosine. This paper demonstrate that using simulation the proposal method is a predominant method to compare with the existing method.