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Double-Angle and Half-Angle Formulas

Double-Angle and Half-Angle Formulas . Section 5.5. Multiple-Angle Formulas. In the previous sections, we used: The Fundamental Identities Sin²x + Cos²x = 1 Sum & Difference Formulas Cos (u – v) = Cos u Cos v + Sin u Sin v Now we will use double angle and half angle formulas.

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Double-Angle and Half-Angle Formulas

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  1. Double-Angle and Half-Angle Formulas Section 5.5

  2. Multiple-Angle Formulas • In the previous sections, we used: • The Fundamental Identities • Sin²x + Cos²x = 1 • Sum & Difference Formulas • Cos (u – v) = Cos u Cos v + Sin u Sin v Now we will use double angle and half angle formulas

  3. Double-Angle Formulas • Double-angle formulas are the formulas used most often:

  4. Double-Angle Formulas • Use the following triangle to find the following: Sin 2θ 2 Cos 2θ θ Tan 2θ 5

  5. Double-Angle Formulas • Use the following triangle to find the following: Sin 2θ = 2Sin θ Cos θ 2 θ 5

  6. Double-Angle Formulas Cos 2θ = 2Cos² θ - 1 2 θ 5

  7. Double-Angle Formulas Tan 2θ 2 θ 5

  8. Double-Angle Formulas • Use the following triangle to find the following: Csc 2θ 1 Sec 2θ θ 4 Cot 2θ

  9. Double-Angle Formulas • General guidelines to follow when the double-angle formulas to solve equations: • Apply the appropriate double-angle formula • Look to factor • Solve the equation using the different strategies involved in solving equations

  10. Double-Angle Formulas • Solve the following equation in the interval [0, 2π) Sin 2x – Cos x = 0 1. Apply the double-angle formula 2 Sin x Cos x – Cos x = 0 2. Look to factor Cos x (2 Sin x – 1) = 0

  11. Double-Angle Formulas Cos x (2 Sin x – 1) = 0 3. Solve the equation Cos x = 0 2 Sin x - 1= 0 Sin x = ½ x x

  12. Double-Angle Formulas • Solve the following equation in the interval [0, 2π) 2 Cos x + Sin 2x = 0 2 Cos x + 2 Sin x Cos x = 0 2 Cos x (1+ Sin x) = 0 2 Cos x = 0 1 + Sin x = 0

  13. Double-Angle Formulas 2 Cos x = 0 1 + Sin x = 0 Cos x = 0 Sin x = -1 x x

  14. Double-Angle Formulas • Solve the following equations for x in the interval [0, 2π) • Sin 2x Sin x = Cos x • Cos 2x + Sin x = 0 x x

  15. Double-Angle Formulas Sin 2x Sin x = Cos x 2 Sin x Cos x Sin x = Cos x 2 Sin²x Cos x – Cos x = 0 Cos x (2 Sin²x – 1) = 0 Cos x = 0 2 Sin²x – 1 = 0 Sin²x = ½ x Sin x = ± ½ x =

  16. Double-Angle Formulas Cos 2x + Sin x = 0 1 – 2Sin² x + Sin x = 0 2Sin² x - Sin x - 1= 0 (2 Sin x + 1) (Sin x – 1) = 0 2 Sin x + 1 = 0 Sin x – 1 = 0 Sin x = ½ Sin x = 1 x = x

  17. Double-Angle and Half-Angle Formulas Section 5.5

  18. Double-Angle Formulas • Evaluating Functions Involving Double Angles Use the given information to find the following: Sin 2x Cos 2x Tan 2x

  19. Double-Angle Formulas 13 12 x -5 Sin 2x = 2Sin x Cos x

  20. Double-Angle Formulas 13 12 x -5 Cos 2x = 2Cos² x - 1

  21. Double-Angle Formulas 13 12 Tan 2x x -5

  22. Double-Angle Formulas • Evaluating Functions Involving Double Angles Use the given information to find the following: Sin 2x Cos 2x Tan 2x

  23. Double-Angle Formulas 8 x -15 17 Sin 2x = 2Sin x Cos x

  24. Double-Angle Formulas 8 x -15 17 Cos 2x = 2Cos² x - 1

  25. Double-Angle Formulas 8 x -15 Tan 2x 17

  26. The next (and final) set of formulas we have are called half-angle formulas. The sign of Sin and Cos depend on what quadrant u/2 is in

  27. Half-Angle Formulas • Use the following triangle to find the six trig functions of θ/2 25 7 θ

  28. Half-Angle Formulas 25 7 θ 24

  29. Half-Angle Formulas 25 7 θ 24

  30. Half-Angle Formulas 25 7 θ 24

  31. Half-Angle Formulas Find the exact value of the Cos 165º. 165º is half of what angle? Cos 165º =

  32. Half-Angle Formulas Find the exact value of the Sin 105º. 105º is half of what angle? Sin 105º =

  33. Half-Angle Formulas Find the exact value of the Tan 15º. 15º is half of what angle? Tan 15º =

  34. Double-Angle and Half-Angle Formulas Section 5.5

  35. Half-Angle Formulas 13 12 x -5

  36. Half-Angle Formulas 13 12 x -5

  37. Half-Angle Formulas 13 12 x -5

  38. Half-Angle Formulas 13 12 x -5

  39. Half-Angle Formulas 5 3 x 4

  40. Half-Angle Formulas 5 3 x 4

  41. Half-Angle Formulas 5 3 x 4

  42. Half-Angle Formulas 5 3 x 4

  43. Half-Angle Formulas • Solving Equations using the half-angle formulas: • Apply the appropriate formula • Use the various methods we have learned to solve equations • Factor • Combine Like Terms • Isolate the Trig Function • Solve the Equation for an Angle(s)

  44. Half-Angle Formulas • Solve the following equation for x in the interval [0, 2π)

  45. Half-Angle Formulas • Solve the following equation for x in the interval [0, 2π)

  46. Half-Angle Formulas

  47. Half-Angle Formulas • Solve the following equation for x in the interval [0, 2π)

  48. Half-Angle Formulas

  49. Because we squared both sides, check your answers!

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