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Homework, Page 366

Homework, Page 366. Find the values of all six trigonometric functions of the angle x . 1. Homework, Page 366. Find the values of all six trigonometric functions of the angle x . 5. Homework, Page 366.

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Homework, Page 366

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  1. Homework, Page 366 Find the values of all six trigonometric functions of the angle x. 1.

  2. Homework, Page 366 Find the values of all six trigonometric functions of the angle x. 5.

  3. Homework, Page 366 Assume that θ is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric functions. 9.

  4. Homework, Page 366 Assume that θ is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric functions. 13.

  5. Homework, Page 366 Assume that θ is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric functions. 17.

  6. Homework, Page 366 Evaluate without using a calculator. 21.

  7. Homework, Page 366 Evaluate using a calculator. Given an exact value. 25.

  8. Homework, Page 366 Evaluate using a calculator, giving answers to three decimal places. 29.

  9. Homework, Page 366 Evaluate using a calculator, giving answers to three decimal places. 33.

  10. Homework, Page 366 Evaluate using a calculator, giving answers to three decimal places. 37.

  11. Homework, Page 366 Without a calculator, find the acute angle θ that satisfies the equation. Give θ in both degrees and radians. 41.

  12. Homework, Page 366 Without a calculator, find the acute angle θ that satisfies the equation. Give θ in both degrees and radians. 45.

  13. Homework, Page 366 Solve for the variable shown. 49.

  14. Homework, Page 366 Solve for the variable shown. 53.

  15. Homework, Page 366 Solve for the variable shown. 57.

  16. Homework, Page 366 61. A guy wire from the top of a radio tower forms a 75º angle with the ground at a 55 ft distance from the foot of the tower. How tall is the tower?

  17. Homework, Page 366 65. A surveyor wanted to measure the length of a lake. Two assistants, A and C, positioned themselves at opposite ends of the lake and the surveyor positioned himself 100 feet perpendicular to the line between the assistants and on the perpendicular line from the assistant C. If the angle between his lines of sight to the two assistants is 79º12‘42“, what is the length of the lake?

  18. Homework, Page 366 69. Which of the following expressions does not represent a real number? a. sin 30º b. tan 45º c. cod 90º d. csc 90º e. sec 90º

  19. Homework, Page 366 73. The table is a simplified trig table. Which column is the values for the sine, the cosine, and the tangent functions? The second column is tangent values, because tangent can be greater than one, the third is sine values, because they are increasing and the fourth column is cosine values because they are decreasing.

  20. 4.3 Trigonometry Extended: The Circular Functions

  21. What you’ll learn about • Trigonometric Functions of Any Angle • Trigonometric Functions of Real Numbers • Periodic Functions • The 16-point unit circle … and why Extending trigonometric functions beyond triangle ratios opens up a new world of applications.

  22. Leading Questions We may substitute any real number n for θ in any trig function and find the value of the function. Cosine is negative in the fourth quadrant. Coterminal angles have the same measure. Quadrantal angles have their terminal sides in the center of the quadrants. The period of a trig function tells us how often it takes on identical values.

  23. Initial Side, Terminal Side

  24. Positive Angle, Negative Angle

  25. Coterminal Angles Two angles in an extended angle-measurement system can have the same initial side and the same terminal side, yet have different measures. Such angles are called coterminal angles.

  26. Example Finding Coterminal Angles

  27. Example Finding Coterminal Angles

  28. Example Evaluating Trig Functions Determined by a Point in Quadrant I

  29. Trigonometric Functions of any Angle

  30. Evaluating Trig Functions of a Nonquadrantal Angle θ Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant. Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ. Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator. Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant. Use the coordinates of point P and the definitions to determine the six trig functions.

  31. Signs of Trigonometric Functions

  32. Reference Angles The acute angle made by the terminal side of an angle and the x-axis is called the reference angle. The absolute value of each trig function is equal to the absolute value of the same trig function of the reference angle in the first quadrant. The sign of the trig function is determined by the quadrant in which the terminal side lies.

  33. Example Evaluating More Trig Functions

  34. Example Using one Trig Ratio to Find the Others

  35. Unit Circle The unit circle is a circle of radius 1 centered at the origin.

  36. Trigonometric Functions of Real Numbers

  37. Periodic Function

  38. The 16-Point Unit Circle

  39. Following Questions Graphs of the sine function may be stretched vertically, but not horizontally. Horizontal stretches of the cosine function are the result of changes in its period. Horizontal translations of the sine function are the result of phase shifts. Sinusoids are functions whose graphs have the shape of the sine curve. Sinusoids may be used to model periodic behavior.

  40. Homework • Homework Assignment #28 • Review Section 4.3 • Page 381, Exercises: 1 – 69 (EOO) • Quiz next time

  41. 4.4 Graphs of Sine and Cosine: Sinusoids

  42. Quick Review

  43. Quick Review Solutions

  44. What you’ll learn about • The Basic Waves Revisited • Sinusoids and Transformations • Modeling Periodic Behavior with Sinusoids … and why Sine and cosine gain added significance when used to model waves and periodic behavior.

  45. Sinusoid

  46. Amplitude of a Sinusoid

  47. Example Finding Amplitude Find the amplitude of each function and use the language of transformations to describe how the graphs are related. (a) (b) (c)

  48. Period of a Sinusoid

  49. Example Finding Period and Frequency Find the period and frequency of each function and use the language of transformations to describe how the graphs are related. (a) (b) (c)

  50. Example Horizontal Stretch or Shrink and Period

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