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Understanding Trigonometric Functions: Unit Circle, Arc Length, and Inverse Functions

Explore the concepts of trigonometric functions using the unit circle, arc length, sectors, and inverse trig functions. Learn about angles, sine, cosine, tangent, radian measure, and negative angles in this educational chapter.

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Understanding Trigonometric Functions: Unit Circle, Arc Length, and Inverse Functions

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  1. Chapter 2 Trigonometric Functions

  2. (0,1) • 2.1 Unit Circle • (x,y) (cos(α) , sin(α)) 1 y α • • • (-1,0) (1,0) (0,0) x sin(α) = y cos(α) = x tan(α) = y/x • (0, -1)

  3. 100° 80° 1 110° 70° 120° 60° 130° .8 50° 140° 40° .6 150° 30° .4 20° 160° .2 10° 170° -1 -.8 -.6 -.4 -.2 .2 .4 .6 .8 1 190° 350° -.2 200° 340° -.4 210° 330° -.6 220° 320° 230° 310° -.8 240° 300° 290° 250° -1 260° 280°

  4. 100° 80° • • 1 110° 70° • • 60° 120° • • 50° 130° .8 • • 140° • • 40° Quadrant II .6 Quadrant I • 30° 150° • Sine + Sine + .4 160° • • 20° Cosine - Cosine + Tangent - Tangent + 170° .2 • • 10° -1 -.8 -.6 -.4 -.2 .2 .4 .6 .8 1 • • 350° 190° Quadrant III Quadrant IV -.2 200° • • 340° Sine - Sine - -.4 Cosine - Cosine + • 330° 210° • Tangent + Tangent - -.6 • 320° 220° • • 310° • 230° -.8 • • 240° 300° • • 250° 290° • • 260° 280° -1

  5. 2.2 Arc Length and Sectors C = πd d (1/7)d

  6. 2.2 Arc Length and Sectors r r 2 2 • r 2 (1/7) r r 2 A = πr 2

  7. 2.2 Arc Length and Sectors α s = 360 πd s α •

  8. 2.2 Arc Length and Sectors α s = 360 πd s 50° • 20 in.

  9. 2.2 Arc Length and Sectors 50 s = 360 40π 200π s = 360 = 50° 1.74 in. • 20 in.

  10. 2.2 Arc Length and Sectors α k = 360 πr 2 k α •

  11. 2.2 Arc Length and Sectors α k = 360 πr 2 45 k = 360 36π k 2 K = 14.14 in. 45° • 6 ft.

  12. 2.3 Radian Measure π rad. 2 2 rad. 1 rad. 3 rad. 0 rad. π rad. 2π rad. 6 rad. 4 rad. 5 rad. 3π rad. 2

  13. π 2 100° 80° 110° 70° 120° 60° 130° 50° 140° 40° 5π π 6 150° 30° 6 20° 160° 10° 170° π 180° 0, 2π 190° 350° 200° 340° 210° 330° 220° 320° 230° 310° 240° 300° 290° 250° 260° 3π 280° 2

  14. 2.4 Inverse Trig Functions and Negative Angles 36.87˚ sin (.6) = _____________ ─ 1

  15. 100° 80° 1 110° 70° 120° 60° 130° .8 50° 140° 40° .6 150° 30° .4 20° 160° .2 10° 170° -1 -.8 -.6 -.4 -.2 .2 .4 .6 .8 1 190° 350° -.2 200° 340° -.4 210° 330° -.6 220° 320° 230° 310° -.8 240° 300° 290° 250° -1 260° 280°

  16. 2.4 Inverse Trig Functions and Negative Angles or 143.13˚ 36.87˚ sin (.6) = ____________________ ─ 1 36.87˚ + 360n 143.13˚ + 360n

  17. 2.4 Inverse Trig Functions and Negative Angles 66.42˚ cos (.4) = ____________________ ─ 1

  18. 100° 80° 1 110° 70° 120° 60° 130° .8 50° 140° 40° .6 150° 30° .4 20° 160° .2 10° 170° -1 -.8 -.6 -.4 -.2 .2 .4 .6 .8 1 190° 350° -.2 200° 340° -.4 210° 330° -.6 220° 320° 230° 310° -.8 240° 300° 290° 250° -1 260° 280°

  19. 2.4 Inverse Trig Functions and Negative Angles or 293.58˚ 66.42˚ cos (.4) = ____________________ ─ 1 66.42˚ + 360n 293.58˚ + 360n

  20. 2.4 Inverse Trig Functions and Negative Angles 68.2˚ tan (2.5) = _____________ ─ 1

  21. 100° 80° 1 110° 70° 120° 60° 130° .8 50° 140° 40° .6 150° 30° .4 20° 160° .2 10° 170° -1 -.8 -.6 -.4 -.2 .2 .4 .6 .8 1 190° 350° -.2 200° 340° -.4 210° 330° -.6 220° 320° 230° 310° -.8 240° 300° 290° 250° -1 260° 280°

  22. 2.4 Inverse Trig Functions and Negative Angles or 248.2˚ 68. 2˚ tan (2.5) = ____________________ ─ 1 68.2˚ + 180n

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