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Trigonometry Fundamentals and Applications

Exploring concepts of the unit circle, trigonometric functions, quadrants, arc length, sectors, radian measure, and inverse trigonometric functions. Includes calculations and examples for sine, cosine, tangent, and negative angles. Understand how to derive values and solve trigonometric equations with detailed explanations.

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Trigonometry Fundamentals and Applications

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

  2. 100° 80° 1 110° 70° 120° 60° 130° 50° .8 140° 40° .6 150° 30° .4 20° 160° .2 10° 170° A D -10 -1 -.8 -.6 -.4 -.2 .2 .4 .6 .8 10 1 -5 5 15 20 190° 350° -.2 10 200° 340° -.4 210° 330° -.6 220° 320° 230° 310° -.8 240° 300° 290° 250° -1 260° 280°

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

  4. 2.2 Arc Length and Sectors C = πd d

  5. 2.2 Arc Length and Sectors r2 r2 • r (1/7) r2 r2 A = πr 2

  6. 2.2 Arc Length and Sectors α 360 πd = s s α •

  7. 2.2 Arc Length and Sectors α 360 πd = s s 50° • 20 in.

  8. 2.2 Arc Length and Sectors 50 s 360 40π = 200π 360 = 1.74 in. s = 50° • 20 in.

  9. 2.2 Arc Length and Sectors α 360 πr = k 2 k α •

  10. 2.2 Arc Length and Sectors α 360 πr = 45 k 360 36π = k 2 k 2 K = 14.14 in. 45° • 6 ft.

  11. 2.3 Radian Measure π 2rad. 2 rad. 1 rad. 3 rad. 0 rad. 2π rad. 6 rad. π rad. 4 rad. 5 rad. 3π 2 rad.

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

  13. 2.4 Inverse Trig Functions and Negative Angles sin (.6) = _____________ 36.87˚ ─ 1

  14. 100° 80° 1 110° 70° 120° 60° 130° 50° .8 140° 40° .6 150° 30° .4 20° 160° .2 10° 170° A D -10 -1 -.8 -.6 -.4 -.2 .2 .4 .6 .8 10 1 -5 5 15 20 190° 350° -.2 10 200° 340° -.4 210° 330° -.6 220° 320° 230° 310° -.8 240° 300° 290° 250° -1 260° 280°

  15. 2.4 Inverse Trig Functions and Negative Angles sin (.6) = ____________________ 36.87˚ or 143.13˚ ─ 1 36.87˚ + 360n 143.13˚ + 360n

  16. 2.4 Inverse Trig Functions and Negative Angles cos (.4) = ____________________ 66.42˚ ─ 1

  17. 100° 80° 1 110° 70° 120° 60° 130° 50° .8 140° 40° .6 150° 30° .4 20° 160° .2 10° 170° A D -10 -1 -.8 -.6 -.4 -.2 .2 .4 .6 .8 10 1 -5 5 15 20 190° 350° -.2 10 200° 340° -.4 210° 330° -.6 220° 320° 230° 310° -.8 240° 300° 290° 250° -1 260° 280°

  18. 2.4 Inverse Trig Functions and Negative Angles cos (.4) = ____________________ 66.42˚ or 293.58˚ ─ 1 66.42˚ + 360n 293.58˚ + 360n

  19. 2.4 Inverse Trig Functions and Negative Angles tan (2.5) = _____________ 68.2˚ ─ 1

  20. 100° 80° 1 110° 70° 120° 60° 130° 50° .8 140° 40° .6 150° 30° .4 20° 160° .2 10° 170° A D -10 -1 -.8 -.6 -.4 -.2 .2 .4 .6 .8 10 1 -5 5 15 20 190° 350° -.2 10 200° 340° -.4 210° 330° -.6 220° 320° 230° 310° -.8 240° 300° 290° 250° -1 260° 280°

  21. 2.4 Inverse Trig Functions and Negative Angles or 248.2˚ tan (2.5) = ____________________ 68. 2˚ ─ 1 68.2˚ + 180n

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