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1.6

1.6. Trigonometric Functions. What you’ll learn about…. Radian Measure Graphs of Trigonometric Functions Periodicity Even and Odd Trigonometric Functions Transformations of Trigonometric Graphs Inverse Trigonometric Functions …and why

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1.6

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  1. 1.6 Trigonometric Functions

  2. What you’ll learn about… • Radian Measure • Graphs of Trigonometric Functions • Periodicity • Even and Odd Trigonometric Functions • Transformations of Trigonometric Graphs • Inverse Trigonometric Functions …and why Trigonometric functions can be used to model periodic behavior and applications such as musical notes.

  3. Radian Measure • The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle.

  4. Radian Measure • An angle of measure θ is placed in standard position at the center of circle of radius r,

  5. Trigonometric Functions of θ

  6. Graphs of Trigonometric Functions • When we graph trigonometric functions in the coordinate plane, we usually denote the independent variable (radians) by x instead of θ .

  7. Angle Convention

  8. Periodic Function, Period

  9. Even and Odd Trigonometric Functions • The graphs of cos x and sec x are even functions because their graphs are symmetric about the y-axis. • The graphs of sin x, csc x, tan x and cot x are odd functions.

  10. Example Even and Odd Trigonometric Functions

  11. Transformations of Trigonometric Graphs • The rules for shifting, stretching, shrinking and reflecting the graph of a function apply to the trigonometric functions. Vertical stretch or shrink Reflection about x-axis Vertical shift Horizontal stretch or shrink Reflection about the y-axis Horizontal shift

  12. Example Transformations of Trigonometric Graphs [-5, 5] by [-4,4]

  13. Inverse Trigonometric Functions • None of the six basic trigonometric functions graphed in Figure 1.42 is one-to-one. These functions do not have inverses. However, in each case, the domain can be restricted to produce a new function that does have an inverse. • The domains and ranges of the inverse trigonometric functions become part of their definitions.

  14. Inverse Trigonometric Functions

  15. Inverse Trigonometric Functions • The graphs of the six inverse trigonometric functions are shown here.

  16. Example Inverse Trigonometric Functions

  17. Quick Quiz Sections 1.4 – 1.6

  18. Quick Quiz Sections 1.4 – 1.6

  19. Quick Quiz Sections 1.4 – 1.6

  20. Quick Quiz Sections 1.4 – 1.6

  21. Quick Quiz Sections 1.4 – 1.6

  22. Quick Quiz Sections 1.4 – 1.6

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