A plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem

We present a plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem. We first show that every triplet of positive real numbers (a, b, c) satisfying a4 + b4 = c4 forms the sides of an acute triangle. The subsequent proof is founded upon the observation that the Pythagorean description of every such triangle expressed through the law of cosines must exactly equal the description of the triangle from the Fermat equation. On the basis of a geometric construction motivated by this observation, we derive a class of polynomials, the roots of which are the sides of these triangles. We show that the polynomials for a given triangle cannot all have rational roots. To the best of our knowledge, the approach offers new geometric and algebraic insight into the irrationality of the roots.

Geographic Information Systems Practice Midwife In Merauke Regency Android-Based Method Using Spherical Law of Cosines

Health is a resource for the daily life of the community. Because without health, a person can not run the activity with the good. To that end, the society can not be separated from the figure of health that play a role in manifesting the life of a healthy society as a midwife. One of the reasons people really need a midwife is the cost of the examination or the cost of labor is affordable. Therefore, it is needed a Geographic Information System Midwife Practice in Merauke Regency Android-Based Method Using Spherical Law of Cosines can help and facilitate user to search, and know the location of the midwife practice and service information available at the place of such practices as well as the closest distance to the user. Geographic Information system Midwife Practice in Merauke Regency Android Based on the design by using Eclipse with the java programming language, and using a MySQL database. The method used to search the nearest distance of the user with the location of the midwife practice using the method of Spherical Law of Cosines, and the method of testing using the method of blackbox and questionnaires. The result of this research is to produce a geographic information system application based on android to handle mapping the location of the midwife practice as well as data information midwife online which is accessible to the public in Merauke Regency.

4. Identities, and more identities

The world of trigonometry is full of identities: some of them extremely useful, others beautiful, and a few that are simply bizarre. ‘Identities, and more identities’ takes a tour of the menagerie of identities, viewing a little from each of these categories. The first two examples are known as triangle identities, because they refer to angles and lengths in a given triangle. The Law of Sines and the Law of Cosines are discussed, along with Mollweide’s formulas, the Law of Tangents, Morrie’s Law, and the introduction of logarithms, which became the preferred computing tool in mathematical astronomy, and then in practical disciplines like surveying and architecture in the early 17th century.

Developing and Applying the Law of Cosines

In history, geometry was founded more as a practical endeavor than a theoretical one. Early developments of the branch portray philosophers' attempts to make sense of their surroundings, including the measurement of distances on earth and in space. Such a link between earth and space sciences and geometry motivated us to develop and implement a multidisciplinary lesson focusing on the conceptual understanding of the law of cosines in the context of astronomy. In our content specific STEAM lesson, the authors aimed to facilitate an understanding of the law of cosines in ninth grade students, and then apply the law in a star map task to find approximate distances between stars. The second part of the lesson also included the use of an instructional technology to support students' work with the star map task. In the conclusion, the authors discuss possible ways to improve the quality of their STEAM education efforts for the given context.

Determination of the junction angle in fluvial channels from georeferenced aerial images from Google Earth Pro and UAV

The junction angles in fluvial channels are determined from complex erosion and deposition processes, resulting from river-flow dynamics, bed and margin morphology, and so on. Knowledge regarding these angles is important in order to better understand the existing conditions in a basin. In this sense, the objective of the present study was to determine the junction angles on fluvial channels, called α, β and γ, applying the law of cosines. Georeferenced Google Earth Pro images and UAV images were used. Then, the values calculated from the georeferenced aerial images were compared with the values calculated from the minimum energy principle. To visualize and understand the obtained angles, the Junction Angles Diagram was used. The obtained result shows that the methodology using georeferenced aerial images have good performance for determining junction angles on fluvial channels.

A Formula to Predict the Magnitude of Achilles Tendon Lengthening Required to Correct Equinus Deformity

Objectives: Achilles tendon lengthening (ATL) is one of the most commonly performed procedures in paediatric orthopaedic surgery. An appropriate adjustment of the amount of ATL is crucial to avoid insufficient or excessive lengthening. However, there is currently no effective method to preoperatively calculate the tendon length needed for equinus deformity correction. Thus, in this study we evaluated the accuracy of a calculation using a mathematical model based on the law of cosines. Methods: A total of 16 feet of 14 patients who were scheduled for ATL surgery due to equinus deformity were included in the study. ATL surgery was performed using a standard Z-plasty technique. Calculation of the amount of ATL using the law of cosines, and assessments of intraoperative lengthening of the tendon, were performed in a double-blind manner. The extent of lengthening resulting from the two methods was then compared. Results: The mean ATL determined intraoperatively was 23.67 ± 8.7 mm, and that obtained using the cosine-based method was 22.49 ± 8.6 mm. Thus, the new method showed excellent statistical agreement with the actual lengthening performed during surgery. Conclusions: The required dimension of ATL can be calculated preoperatively using the mathematical formula presented here. The advantages of this approach are that it allows accurate tendon lengthening and reduces the size of the surgical incision.