The hyperbolic and circular trigonometric functions exhibit periodicity under repeated differentiation. The hyperbolic sine and cosine functions, whilst distinct from their first derivatives, are each equal to their own second derivative.... more
The hyperbolic and circular trigonometric functions exhibit periodicity under repeated differentiation. The hyperbolic sine and cosine functions, whilst distinct from their first derivatives, are each equal to their own second derivative. Similarly, the circular sine and cosine functions are each equal to their own fourth derivative. Very little appears in the literature about functions of a real variable that are equal to their own third derivative. The purpose of this note is to define three such functions and to investigate their properties.
In 2016, the Mathematical Society of the Philippines (MSP) testified that among Filipino learners of ages 14-18, 85% experienced the dilemma on dealing with the evaluation of special angles in Trigonometry wherein 45% was from public high... more
In 2016, the Mathematical Society of the Philippines (MSP) testified that among Filipino learners of ages 14-18, 85% experienced the dilemma on dealing with the evaluation of special angles in Trigonometry wherein 45% was from public high schools. This study aimed to determine the potential of a visual mathematical hand mnemonic tactic (VMHMT) as fast and accurate classroom mathematical mnemonic strategy in evaluating special angles in Trigonometry. A 20-item multiple choice type test with 50-minute time allotment. Reliability of the test items was also measured via Pearson's. The test was administered three times with 7-day interval for each strategy with uniform starting time. ANOVA results showed no significant difference among the three manual strategies in terms of accuracy of answers (p=0.373) which means that the accuracy of the VMHMT is comparable with that of the other existing manual strategies. Meanwhile, significant difference was obtained in terms of speed of calculations (p=0.000) wherein the use of unit circle rendered longest time for calculations (M=21.234) while the VMHMT (M=14.70) and table of trigonometric ratios (M=14.23) appeared to be very fast. Learners find the strategy more relevant and experiential which could make a more lifelong learning.
Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of... more
Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form (1−αx2)eβx2 for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval (0, π), while our main result for the cosine function is a double inequality holding on the interval (0, π/2). Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l’Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given.
Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of... more
Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form (1 − αx2)eβx2 for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval (0, π), while our main result for the cosine function is a double inequality holding on the interval (0, π/2). Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l’Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given
The objective of this study is to compare the performance of Applied Sciences students’ in a Pre and Post Basic Differentiation and Integration Test during their second year of study at the Faculty of Applied Sciences, Universiti... more
The objective of this study is to compare the performance of Applied Sciences students’ in a Pre and Post Basic Differentiation and Integration Test during their second year of study at the Faculty of Applied Sciences, Universiti Teknologi MARA, Perak Branch, Tapah Campus. This paper focuses on the students’ understanding in basic differentiation and integration for trigonometric functions. A total of 95 Semester 3 students of session 2 2017/2018 from the Department of Applied Sciences are chosen to answer the test. The pre-test was held in week 10, while the post-test was conducted in week 13 of the semester. In pre-test, the students need to answer the questions based on traditional method while in post-test, the EzTrigo Mnemonic Diagram is introduced and the students applied the technique in order to answer the questions. The results showed that the students’ performance in post-test was better compared to that pre-test
Making connections between the representations of trigonometric functions and an interpretation of graphs of the functions are major challenges to many students. This study explores the effectiveness of the GeoGebra on grade 12 students'... more
Making connections between the representations of trigonometric functions and an interpretation of graphs of the functions are major challenges to many students. This study explores the effectiveness of the GeoGebra on grade 12 students' success in making connections between the representations of trigonometric functions and the interpretation of graphs. A nonequivalent control-group pre-test post-test quasi-experimental design was used. The sample of the study consisted of sixtyone grade 12 students from two schools. The results showed that there was a statistically significant difference between the mean achievements of the experimental group and the control group on making connections between representations of trigonometric functions, and on analyses and interpretations of representations of trigonometric functions, in favour of the experimental group. This study extends the findings of previous studies on the effectiveness of dynamic mathematics software on students' learning of representations and interpretation of graphs of trigonometric functions.
A generalized convolution with a weight-function for the Fourier cosine, Mellin and Fourier sine integral transforms is introduced. Its properties are studied, the application to solving a system of integral equations is outlined.
The objective of this study is to compare the performance of Applied Sciences students’ in a Pre and Post Basic Differentiation and Integration Test during their second year of study at the Faculty of Applied Sciences, Universiti... more
The objective of this study is to compare the performance of Applied Sciences students’ in a Pre and Post Basic Differentiation and Integration Test during their second year of study at the Faculty of Applied Sciences, Universiti Teknologi MARA, Perak Branch, Tapah Campus. This paper focuses on the students’ understanding in basic differentiation and integration for trigonometric functions. A total of 95 Semester 3 students of session 2 2017/2018 from the Department of Applied Sciences are chosen to answer the test. The pre-test was held in week 10, while the post-test was conducted in week 13 of the semester. In pre-test, the students need to answer the questions based on traditional method while in post-test, the EzTrigo Mnemonic Diagram is introduced and the students applied the technique in order to answer the questions. The results showed that the students’ performance in post-test was better compared to that pre-test.
Here, the miscellaneous soliton solutions of the generalized nonlinear Schrödinger equation are considered that describe the model of few-cycle pulse propagation in metamaterials with parabolic law of nonlinearity. The novel analytical... more
Here, the miscellaneous soliton solutions of the generalized nonlinear Schrödinger equation are considered that describe the model of few-cycle pulse propagation in metamaterials with parabolic law of nonlinearity. The novel analytical wave solutions to the mentioned nonlinear equation in the sense of the nonlinear ordinary differential transform equation are obtained. The techniques are the improved exp − Γ ϖ function method and the improved simple equation method. The nonlinear ordinary transform to concern the generalized Schrodinger equation to convert it for a solvable integer-order differential equation is used. After the successful implementation of the presented methods, the exact solitary wave solutions in the form of trigonometric, rational, and hyperbolic functions are obtained. Hence, the presented methods are relatable and efficient to solve nonlinear problems in mathematical physics.
Making connections between the representations of trigonometric functions and an interpretation of graphs of the functions are major challenges to many students. This study explores the effectiveness of the GeoGebra on grade 12... more
Making connections between the representations of trigonometric functions and an interpretation of graphs of the functions are major challenges to many students. This study explores the effectiveness of the GeoGebra on grade 12 students' success in making connections between the representations of trigonometric functions and the interpretation of graphs. A nonequivalent control-group pre-test post-test quasi-experimental design was used. The sample of the study consisted of sixtyone grade 12 students from two schools. The results showed that there was a statistically significant difference between the mean achievements of the experimental group and the control group on making connections between representations of trigonometric functions, and on analyses and interpretations of representations of trigonometric functions, in favour of the experimental group. This study extends the findings of previous studies on the effectiveness of dynamic mathematics software on students' le...
