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Trigonometric Functions and Periodic Properties

Discover the concepts of trigonometric functions, periodicity, and key features such as amplitude and midline. Learn about the graphs of sin, cos, and tan functions, along with transformations and analysis.

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Trigonometric Functions and Periodic Properties

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  1. Trigonometric Functions

  2. Information

  3. Periodic functions A function with a pattern that repeats at regular intervals is called a periodic function. The length of the interval over which a periodic function repeats is called the period of the function. What is the period of the function shown above? period = 4

  4. Trigonometric functions of real numbers The trigonometric functions are common periodic functions. For any real numberθ, the trigonometric functions are defined: sinθ = y/r cosθ = x/r tanθ = y/x where x and y are the coordinates of any point P(x, y) on the terminal ray of an angle of θradians when it is in standard position, and r is the distance from the origin to P. r θ In particular, if P is on the unit circle: sinθ = y cosθ = x tanθ = y/x

  5. The graph of sinθ

  6. Sine parent function: f(x) = sinx (– ∞, ∞) domain: range: [–1, 1] asymptotes: none period: 2π roots: x = nπ maxima: x = π⁄2 +2nπ minima: x = –π⁄2 +2nπ for every integern Is sinx an even or odd function?

  7. The graph of cosθ

  8. Cosine parent function: f(x) = cosx (– ∞, ∞) domain: [–1, 1] range: asymptotes: none period: 2π roots: x = π⁄2 +nπ maxima: x = 2nπ minima: x = π+2nπ for every integern Is cosx an even or odd function?

  9. The graph of tanθ

  10. Tangent parent function: f(x) = tanx (nπ–π⁄2 , nπ+π⁄2 ) domain: (– ∞, ∞) range: period: π roots: x = nπ horizontal asymptotes: none vertical asymptotes: x = π⁄2 +nπ for every integern Is tanx an even or odd function?

  11. Amplitude and midline A key feature of a periodic function is its amplitude. The amplitude of a periodic function is half the difference between the minimum and maximum y values, (maxy – miny)/2. amplitude midline The midline of a periodic function is the line at the midpoint of the range. y = miny + (maxy – miny)/2 What is the amplitude of the graph shown? What is the equation of its midline?

  12. Trigonometric functions

  13. Transforming trigonometric graphs

  14. Transformations and key features

  15. Analyzing functions Graph f(x)= –3cos2x. 1) Find the domain, range, period and amplitude. 2) Find the x-values where the minimum and maximum values occur in the interval [0, 2π]. 3) State the zeros in the interval [0, 2π]. (– ∞, ∞) 1) domain: range: [– 3, 3] period: π amplitude: 3 2) minima: x = 0, π, 2π maxima: x = π/2, 3π/2 3) zeroes: x =π/4, 3π/4, 5π/4, 7π/4

  16. General form

  17. Match the graph to the equation

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