Matematika - Persamaan Trigonometri Sederhana

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Matematika - Persamaan Trigonometri Sederhana

  1. 1. Trigonometry SIMPLE EQUATION
  2. 2. Trigonomentri simpleequation is an equation that contains thecomparison trigonomentri
  3. 3. In general, to solvetrigonometry is used in the following formula:
  4. 4. 1. sin x = sin α x = α + k.360⁰ atau x = (180⁰ - α) + k.360⁰
  5. 5. 2. cos x = cos αx = α + k.360⁰ atau x = - α + k.360⁰
  6. 6. 3. tan x = tan α x = α + k.180⁰
  7. 7. For angle in units of radians, in use the following formula:
  8. 8. 1. sin x = sin α x = α + k.2π or x = (π – α) + k.2π
  9. 9. 2. cos x = sin αx = α + k.2π or x = -α + k.2π
  10. 10. 3. tan x = tan α x = α + k.π
  11. 11. Example - exampleproblems trigonometric equations:
  12. 12.  Determine the set of equations following the completion of the interval 0 ≤ x ≤ 2πa. Sin x = ½ √3b. Tan x = √3
  13. 13. Answer :a. Sin x = ½√3 = sin (π/3 + k. 2π) x = π/3 + k. 2π for k = 0 x = π/3
  14. 14. ORsin x = ½ √3 = sin ( π – π/3 + k.2π) x = 2π/3 + k . 2π For k = 0 x = 2π/3
  15. 15. Thus, the solution set = {π / 3, 2π / 3}
  16. 16. tan x = √3 = tan (π/3 + k. π) x = π/3 + k . πFor k = 0 x = π/3 + k.Π k=1 x = 4π/3
  17. 17. so, the solution set = {Π / 3, 4π / 3}
  18. 18. 2. Determine the set of completion of the equation cos (3x - 45 ⁰) = - ½ √ 2, for 0 ⁰ ≤ x ≤ 360 ⁰.
  19. 19. Answer : Cos (3x – 45⁰) = -½√2 Cos (3x – 45⁰) = cos 135⁰ 3x – 45⁰ = 135⁰ + k. 360⁰ 3x = 180⁰ + k. 360⁰ x = 60⁰ + k. 120⁰ x = 60⁰ , 180⁰ , 300⁰
  20. 20. OR3x – 45⁰ = -135 + k . 360⁰ 3x = -90⁰ + k . 360⁰ x = -30⁰ + k . 120⁰ x = 90⁰ , 210⁰ , 330⁰
  21. 21. Thus, the solution set = {60 ⁰, 90 ⁰, 180 ⁰, 210 ⁰, 300 ⁰, 330 ⁰}
  22. 22. The basic technique is to solvetrigonometric equations using trigidentities and algebra techniques totransform a trigonometric equation intosimpler forms.
  23. 23. example:Determine the set ofcompletion of sin x = sin 70°,0° <x <360 °
  24. 24. Answer : x = 70° + k.360° k = 0 ==> x = 70° atau x = (180 - 70) + k.360° ==> x=110° + k.360° k = 0 ==> x = 110° Jadi, Hp = {70°, 110°}
  25. 25. Determine the set ofcompletion of cos x = cos 24in the interval 0 ° <x <360 °
  26. 26. Answer : x = 24° + k.360° k = 0 , x = 24° OR x = -24° + k.360° k = 1 , x = -24° + 360° = 336° Thus, Hp = {24°, 336°}
  27. 27. Determine the set of completion of tan x = tan 56 °, in the interval 0 ° <x <360 °
  28. 28. Answer: x = 56 ° + ° k.180 k = 0 ==> x = 56 ° k = 1 ==> x = 56 ° + 180 ° = 236 ° Thus, the solution set is {52 °, 236 °}
  29. 29. THANK YOU

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