This book is the result of the authors' work in fractional calculus, and more particularly, in functions for the solutions of fractional differential equations, which is fostered in the behavior of generalized exponential functions. Classical trigonometry plays a very important role relative to integer order calculus, and together with the common exponential function, provides solutions for linear differential equations. The authors discuss how fractional trigonometry plays an analogous role relative to the fractional calculus by providing solutions to linear fractional differential equations. The importance of the classical trigonometry goes far beyond the solutions of triangles. Its use in Fourier integrals, Fourier series, signal processing, harmonic analysis, and more provide great motivation for the development of a fractional trigonometry to expand application to the fractional calculus domain. The book begins with an introductory chapter that offers insight to the fundamentals of fractional calculus, and topical coverage is then organized in two main parts. Part One consists of Chapters 2 through 11, which develop the definitions and theories of fractional exponentials and fractional trigonometry. Part Two consists of Chapters 12 to 19, which provide insight into various areas of potential application within the sciences. Chapter coverage includes: Introduction; The Fractional Exponential Function via the Fundamental Fractional Differential Equation; The Generalized Exponential Function; R-Function Relationships; The Fractional Hyperboletry; The R_{1} Fractional Trigonometry; The R_{2} Fractional Trigonometry; The R_{3} Trigonometric Functions; The Fractional Meta-Trigonometry; The Ratio and Reciprocal Functions; Further Generalized Fractional Trigonometries; The Solution of Linear Fractional Differential Equations based on the Fractional Trigonometry; Fractional Dynamics and Fractional Systems; Numerical Issues and Approximations in the Fractional Trigonometry; The Fractional Spiral Functions; Fractional Oscillators; Shell Morphology and Growth; Mathematical Classification of the Spiral and Ring Galaxy Morphologies; and Hurricanes, Tornados, and Whirlpools.