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Special Functions
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
3. Law of sines: asinA=bsinB=csinC. $ \frac{a}{{ {\text{sin }}A}} = \frac{b}{{ {\text{sin }}B}} = \frac{c}{{ {\text{sin }}C}}. $
A novel plastic failure model of prestressed anchor cables for upper bound stability analyses of rotational reinforced slopes
Published in European Journal of Environmental and Civil Engineering, 2023
Ze-Hang Qian, Jin-Feng Zou, Xing-Xing Wei
Given that the anchor head perfectly attaches to the slope surface, the movement of the anchor head is compatible with that of the slopes. As shown in Figure 8, an earth slope is subjected to a rotational displacement which drives the anchor head to move from to Considering the triangle according to the law of sines, the moving distance of the anchor heads with a rotating angle (equal to the rotational angle of slopes) can be computed using the following equation:
An empirical study of mathematical creativity and students’ opinions on multiple solution tasks
Published in International Journal of Mathematical Education in Science and Technology, 2022
Ljerka Jukić Matić, Josip Sliško
To justify the use of our chosen problem, we first show that it can be solved in multiple ways. Following Levav-Waynberg and Leikin (2013), we constructed an expert solution space for the given MST. We found nine solution methods that might be accessible to students at the end of high school in the Croatian education system. Some solution methods used the same concept, for example, area of a triangle, but the triangle area could be found using two completely different paths. Solution methods from trigonometry may employ different theorems, properties and definitions. All of the solution methods can be regarded as conventional in the context of the Croatian mathematics curriculum, except for solution methods 7 and 9. Solution method 7 uses trigonometric identities, the law of cosines and the law of sines. Those theorems can be utilized in any triangle, but they are rarely used in the right triangle as described in Figure 2. Solution method 9 uses coordinate geometry. Coordinate geometry is part of the Croatian mathematics curriculum, but this solution method can be regarded as unconventional because the task does not suggest coordinate geometry in any way, and the mathematics curriculum prescribes coordinate geometry in different contexts (MZO, 2019). We asked two university mathematics and high school mathematics teachers to evaluate the expert solution space: which solution methods were equivalent, which were not included but could be, or which methods were not possible for Croatian high school students. The comments of the raters were also considered and this process resulted in nine solution methods with refined descriptions (Figure 2).
Ultimate pullout capacity of single vertical plate anchors in sand
Published in Marine Georesources & Geotechnology, 2021
Peizhi Zhuang, Hongya Yue, Xiuguang Song, Renjuan Sun, Jianqing Wu, Yanhua Guan
The radius of any point (i.e., G1) on the logarithmic spiral BC can be expressed as Thus, the coordinates of G1 in the Cartesian space defined in Figure 6 are: where θ1 represents the value of ∠BO1G1. r1 is the length of line O1B, which can be calculated based on the law of sines as: