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4.3

4.3. Demana, Waits, Foley, Kennedy. Trigonometric Functions of Any Angle. 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

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4.3

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  1. 4.3 Demana, Waits, Foley, Kennedy Trigonometric Functions of Any Angle

  2. 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.

  3. Initial Side, Terminal Side

  4. Positive Angle, Negative Angle

  5. 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.

  6. Example: Finding Coterminal Angles

  7. Solution

  8. Example: Finding Coterminal Angles

  9. Solution

  10. Example: Evaluating Trig Functions Determined by a Point in QI

  11. Solution

  12. Trigonometric Functions of any Angle

  13. Reference Angles

  14. 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.

  15. Evaluating Trig Functions of a Nonquadrantal Angle θ 4. 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. 5. Use the coordinates of point P and the definitions to determine the six trig functions.

  16. Example: Evaluating More Trig Functions

  17. Solution

  18. Example: Using One Trig Ratio to Find the Others

  19. Solution

  20. Solution Continued

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

  22. Trigonometric Functions of Real Numbers

  23. Periodic Function

  24. The 16-Point Unit Circle

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