1 / 36

Circular Trigonometric Functions

Circular Trigonometric Functions. Special Angles. *Special Angles. 30 ° , 45°, and 60° → common reference angles Memorize their trigonometric functions. Use the Pythagorean Theorem; triangles below. 60 °. 45 °. 2. 1. 1. 45 °. 30 °. 1. * Special Angles. * Special Angles.

zev
Download Presentation

Circular Trigonometric Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Circular Trigonometric Functions

  2. Special Angles

  3. *Special Angles 30°, 45°, and 60° → common reference angles Memorize their trigonometric functions. Use the Pythagorean Theorem; triangles below. 60° 45° 2 1 1 45° 30° 1

  4. *Special Angles

  5. *Special Angles

  6. Find trig functions of 300° without calculator. Reference angle is 60°[360° - 300°]; IV quadrant sin 300° = - sin 60° cos 300° = cos 60° tan 300° = - tan 60° csc 300° = - csc 60° sec 300° = sec 60° cot 300° = - cot 60° 60° 300° Use special angle chart.

  7. sin 300° =- sin 60° = - 0.8660 = cos 300° = cos 60° = 0.5000 = 1/ 2 tan 300° = - tan 60° = -1.732 = csc 300° = - csc 60° = -1.155 = sec 300° = sec 60° = 2.000 = 2/1 cot 300° = - cot 60° = - 0.5774 =

  8. Quadrant Angles

  9. *Quadrant Angles • Reference angles cannot be drawn for • quadrant angles 0°, 90°, 180°, and 270° • Determined from the unit circle; r = 1 • Coordinates of points (x, y) • correspond to (cos θ, sin θ)

  10. *Quadrant Angles 90° (0,1) → (cos θ, sin θ) r = 1 180° (-1,0) 0° (1,0) 270° (0,-1)

  11. *Quadrant Angles

  12. Find trig functions for - 90°. Reference angle is (360° - 90°) → 270° sin 270° = -1 cos 270° = 0 tan 270° undefined csc 270° = -1 sec 270° undefined cot 270° = 0 -90° Use quadrant angle chart. 270°

  13. Coterminal Angles

  14. *Coterminal Angle The angle between 0º and 360º having the same terminal side as a given angle. Ex. 405º- 360º = coterminal angle 45º θ1 = 405º θ2 = 45º

  15. *Coterminal Angles Used with anglesgreater than 360°, or angles less than 0°. • Example • cos 900° = • cos (900° - 720°) = • cos 180° = -1 • (See quadrant angles chart)

  16. Example • tan (-135° ) = • tan (360° -135°) = • tan 225° = • LOOK→ tan (225° - 180°) • tan 45° = 1 • (See special angles chart)

  17. Find the value of sec 7π / 4 • Convert from radian to degrees: • sec [(7π/ 4)(180/ π)] = sec 315°

  18. Angle in IV quadrant: sec →positive • sec (360° - 315°) = sec 45° • = 1 /(cos 45°) = √2 = 1.414 • Look how this problem was worked • in previous lesson.

  19. Practice

  20. Express as a function of the reference angle and find the value. tan 210° sec 120 °

  21. Express as a function of the reference angle and find the value. sin (- 330°) csc 225°

  22. Express as a function of the reference angle and find the value. cot (9π/2) cos (-5π)

  23. Inverse Trigonometric Functions

  24. Inverse Trig Functions • Used to find the angle • when two sides of right triangle are known... • or if its trigonometric functions are known Notation used: Read: “angle whose sine is …” Also,

  25. Inverse trig functions have only one principal value of the angle defined for each value of x: 0° <arccos< 180° -90° <arcsin< 90° -90° <arctan< 90°

  26. Example: Given tan θ = -1.600, find θ to the nearest 0.1° for 0° <θ< 360° • Tan is negative in II & IV quadrants

  27. θ = 180° - 58.0° = 122° II θ = 360° - 58.0° = 302° IV Note: On the calculator entering results in -58.0°

  28. More Practice

  29. Given sin θ = 0.3843, find θ to the nearest 0.1° for 0° <θ< 360°

  30. Given cos θ = - 0.0157, find θ to the nearest 0.1° for 0° <θ< 360°

  31. Given sec θ = 1.553 where sin θ < 0, find θ to the nearest 0.1° for 0° <θ< 360°

  32. Given the terminal side of θ passes through point (2, -1), find θthe nearest 0.1° for 0° <θ< 360°

  33. Application

  34. The voltage of ordinary house current is • expressed asV = 170 sin2πft , where • f = frequency = 60 Hzand t = time in seconds. • Find the angle2πft in radians when • V = 120 volts and 0 <2πft <2π

  35. Find t when V = 120 volts

  36. The angleβof a laser beam is expressed as: where w = width of the beam (the diameter) and d = distance from the source. Findβif w = 1.00m and d = 1000m.

More Related