A periodic function is a function that repeats its values at regular intervals. This means that the function's values follow a pattern that is reproduced at fixed periods or intervals. Periodic functions are an important concept in various areas of mathematics, including trigonometry, Fourier analysis, and the study of waves and oscillations.
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Periodic functions are essential in the study of the other trigonometric functions, as they are used to define and graph these functions.
Right triangle trigonometry relies on periodic functions, as the trigonometric ratios (sine, cosine, and tangent) are periodic functions of the angle.
The graphs of the other trigonometric functions (secant, cosecant, and cotangent) are also periodic functions.
Periodic functions play a crucial role in the derivation of sum and difference identities, double-angle, half-angle, and reduction formulas, as well as sum-to-product and product-to-sum formulas.
Polar coordinates, which are used to represent points in a plane, are defined using periodic functions, specifically the trigonometric functions sine and cosine.
Review Questions
Explain how periodic functions are used in the study of the other trigonometric functions (5.3 and 5.4)
Periodic functions are fundamental to the definition and graphing of the other trigonometric functions, such as secant, cosecant, and cotangent. These functions are periodic, meaning they repeat their values at regular intervals, just like the primary trigonometric functions of sine, cosine, and tangent. Understanding the periodic nature of these functions is crucial for analyzing their properties, transformations, and applications in right triangle trigonometry.
Describe the role of periodic functions in the graphs of the other trigonometric functions (6.2)
The periodic nature of functions like secant, cosecant, and cotangent is directly reflected in the shapes of their graphs. These graphs exhibit a repeating pattern, with the function values cycling through a specific range at regular intervals. Analyzing the period, amplitude, and other characteristics of these periodic graphs is essential for understanding the behavior and properties of the other trigonometric functions.
Discuss how periodic functions are used in the derivation of trigonometric identities and formulas (7.2, 7.3, and 7.4)
$$\begin{align*}\sin(x + y) &= \sin(x)\cos(y) + \cos(x)\sin(y) \\ \cos(x + y) &= \cos(x)\cos(y) - \sin(x)\sin(y) \\ \tan(2x) &= \frac{2\tan(x)}{1 - \tan^2(x)} \\ \sin(A + B) &= \sin(A)\cos(B) + \cos(A)\sin(B)\end{align*}$$ The periodic nature of trigonometric functions, such as sine and cosine, is crucial in the derivation of important identities and formulas, including sum and difference identities, double-angle formulas, and sum-to-product/product-to-sum transformations. These periodic properties allow for the manipulation and simplification of trigonometric expressions, which is essential in various applications of trigonometry.
The period of a periodic function is the smallest positive value of the independent variable for which the function repeats its values. It represents the interval over which the function's pattern is repeated.
The amplitude of a periodic function is the maximum absolute value of the function's range. It represents the distance between the function's maximum and minimum values.
The frequency of a periodic function is the number of complete cycles or periods that occur in a unit of time. It is the reciprocal of the function's period.