Proof Without Words: Lagrange’s Trigonometric Identity

The rapid development of mathematics is largely due to the creation of complex numbers. Although complex numbers are just strange and seemingly meaningless notations for most people, complex numbers play important roles in engineering fields. A better understanding of complex numbers' geometric and algebraic structure enables us to study complex analysis better. Complex numbers convince scientists that our world is magical, full of wonderful insights, and even miraculous. In this paper, I first review several basic properties of complex numbers. The set of complex numbers is a group under the operation of the multiplication and under the operation of addition. Then I visualize complex numbers by building a bijective connection between complex numbers and points on the complex plane. I also give several alternative forms of complex numbers, namely the trigonometric and general forms. By invoking the arithmetic properties of complex numbers, I prove two trigonometric identities.

The engineer is always learning procedures and techniques that make the solving of certain types of problems easier. These techniques often use simplification procedures that not only simplify the analysis, but also transform the problem so that it can be solved using familiar techniques. A good example of this is the solution of electrical circuits where the sources are sinusoidal. The set of integral-differential equations describing the circuit is transformed into a set of simultaneous algebraic equations involving complex numbers. The writing and solving of these equations deal with techniques that are already a familiar part of the engineer's bag-of-tricks. The circuit is transformed into what is called the “frequency domain,” and the emphasis is transformed into one of learning how to deal with complex numbers and complex functions.

Trigonometric Identities via Complex Numbers

We generalize the super-stability result for cosine functional equation mappings with values in the field of complex numbers to the case of an arbitrary Hilbert space with the Hadamard product. On the other hand, by an example we prove that this generalization is not true for sine functional equation. Mathematics subject classification (2010): Primary: 39B72, 39B82; secondary: 46E40.

A text for students preparing for their A-level examinations in mathematics

A colleague who teaches high school and college algebra, precalculus, and calculus asked us for help. Her high school students wanted to know whether complex algebra is “good for something”. Could we help answer them? She had introduced complex numbers, including drill in arithmetic and algebraic manipulations, and had covered deMoivre's theorem, e iθ = cos(θ) + i sin(θ). We didn't know of any good enrichment materials to help, and so developed and presented to her classes (one in precalculus and one in calculus) the ideas we share with you here. The first example is an extension of “guess and check” to include variables, rather than numbers: the quadratic equation, az 2 + bz + c = 0, is solved by assuming a complex solution of the form z = x+iy. Other examples explore the use of deMoivre's theorem in deriving trigonometric identities, such as sin(2θ) = 2sin(θ) cos(θ). The students seemed to enjoy this material, and felt that it answered their question. An additional message was that a difficult (or intractable) problem can be made straightforward (or at least do-able) by transforming the problem to an equivalent problem in a different area of mathematics.

Proving trigonometric identities is normally approached geometrically, where one constructs a diagram containing useful ratios, to reveal the required answer using logic. In this chapter I show how complex numbers provide an algebraic way of proving trigonometric identities.

Right Triangle Ratios. Trigonometric Functions. Graphing Trigonometric Functions. Identities. Inverse Trigonometric Functions Trigonometric Equations and Inequalities. Additional Topics: Triangles and Vectors. Polar Coordinates Complex Numbers. Appendix A: Comments on Numbers. Appendix B: Functions and Inverse Functions. Appendix C: Plane Geometry: Some Useful Facts. Selected Answers. Index.

In recent years, the classic method of Coordinate Rotation by Digital Computer (CORDIC) arithmetic has been widely implemented as part of the computational requirements of the well known QR-RLS (Recursive Least Squares) algorithm. In order to operate Givens rotation on a complex number value system, double angle complex rotation (DACR) was adopted to simplify the computational requirement of Complex Givens Rotation. This paper presents a new architecture of high speed CORDIC based single Processor Element (PE) that can be used to accomplish the complex value QR update based RLS. The implementation results on Xilinx FPGA implementaton demonstrates that the proposed structure results in a lower latency and lower cost.

This paper discusses the real and complex numbers of binomial coefficients in combinatorial geometric series. The coefficient for each term in combinatorial geometric series refers to a binomial coefficient. This idea can enable the scientific researchers to solve the real life problems.