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Understanding the Unit Circle: Radian Measures and Trigonometric Functions

This guide explores the concept of the unit circle with a radius of 1 and its importance in trigonometry. Key topics include the foundational triangles (30º, 45º, and 60º) and their relationships within the unit circle, the definition of radian measures, and how they relate to arc lengths. The text covers the six trigonometric functions, the existence of certain functions at specific angles, and the periodic nature of these functions. Practice problems are included to reinforce learning on pages 278-279.

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Understanding the Unit Circle: Radian Measures and Trigonometric Functions

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  1. 4.2 Trigonometric Function: The Unit circle

  2. The Unit Circle A circle with radius of 1 Equation x2 + y2 = 1

  3. The Unit Circle with Radian Measures

  4. Do you remember 30º, 60º, 90º triangles? Now they are really! Important

  5. Do you remember 30º, 60º, 90º triangles? Now they are really! Important Even more important Let 2a = 1

  6. Do you remember 30º, 60º, 90º triangles? Let 2a = 1

  7. Do you remember 30º, 60º, 90º triangles?

  8. Do you remember 45º, 45º, 90º triangles? When the hypotenuse is 1 The legs are

  9. Some common radian measurements These are the Degree expressed in Radians

  10. The Unit Circle: Radian Measures and Coordinates

  11. The Six Trig functions

  12. Why does the book use “t” for an angle? Since Radian measurement are lengths of an arc of the unit circle, it is written as if the angle was on a number line. Where the distance is “t’ from zero. Later when we graph Trig functions it just works better.

  13. Lets find the Trig functions if Think where this angle is on the unit circle.

  14. Find the Trig functions of Think where this angle is on the unit circle.

  15. How about

  16. There are times when Tan or Cot does not exist. At what angles would this happen?

  17. There are times when Tan or Cot does not exist.

  18. If think of the domain of the trig functions, there are some limits. Look at the unit circle. If x goes with Cos, then what are the possible of Cos? It is the same with Sin?

  19. Definition of a Periodic Function A function “f” is periodic if there exist a positive real number “ c” such that f(t + c) = f(t) for all values of “t”. The smallest “c” is called the period.

  20. Even Function ( Trig. ) Cos (- t) = Cos (t) and Sec( -t) = Sec (t) Also Sin(-t) = -sin (t) and Csc (-t) = - Csc (t) Tan(-t) = -Tan (t) and Cot(-t) = - Cot (t)

  21. Homework Page 278- 279 # 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 48, 52, 59, 68

  22. Homework Page 278- 279 # 2, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 49, 58, 61

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