Trigonometri SMA Kelas XI IPA Sem 1

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Trigonometri SMA Kelas XI IPA Sem 1

  1. 1. TRIGONOMETRI http://meetabied.wordpress.com
  2. 2. Setelah menyaksikan tayangan ini anda dapat Menyelesaikan soal-soal yang berkaitan dengan jumlah dan selisih sudut serta sudut rangkap http://meetabied.wordpress.com
  3. 3. Rumus jumlah dan selisih dua sudut sin(  +  ) = sin  .cos  + cos  .sin  sin(  -  ) = sin  .cos  - cos  .sin  http://meetabied.wordpress.com
  4. 4. 1. Sin 75 o = …. Bahasan: sin(  +  ) = sin  .cos  + cos  .sin  sin75 0 = sin(45 0 + 30 0 ) = sin45 0 cos30 0 + cos45 0 sin30 0 = ½√2.½√3 + ½√2.½ = ¼√6 + ¼√2 = ¼√2(√2 + 1) http://meetabied.wordpress.com
  5. 5. 2. Diketahui sin A = cos B = A dan B adalah sudut-sudut lancip sin(A – B) =…. Bahasan: sin(A – B)= sinAcosB – cosAsinB sinA = cosA = ? ? 3 5 4 cos B = sin B = 7 25 24 http://meetabied.wordpress.com A B
  6. 6. sin A =  cos A = cos B =  sin B = sin(A – B) =…. = sinAcosB – cosAsinB = x - x = = http://meetabied.wordpress.com
  7. 7. Rumus jumlah dan selisih dua sudut cos(  +  ) = cos  cos  - sin  sin  cos(  -  ) = cos  cos  + sin  sin  http://meetabied.wordpress.com
  8. 8. 1. Bahasan: cos  cos  + sin  sin  = cos(  -  ) = = = http://meetabied.wordpress.com
  9. 9. 2. a. –sin a .sin b b. cos a .cos b c. sin a .sin b d. 1 – tan a .tan b e. 1 + tan a .tan b http://meetabied.wordpress.com
  10. 10. = = 1 – tan a .tan b  jawab d http://meetabied.wordpress.com
  11. 11. 3. Tentukan nilai cos56 ° + sin56°.tan28° Bahasan: cos56° + sin56°.tan28° = cos56° + sin56°. = cos56° + http://meetabied.wordpress.com
  12. 12. = cos56° + = = = Jadi, Nilai cos56° + sin56°.tan28° = 1 = 1 http://meetabied.wordpress.com
  13. 13. 4. Pada suatu segitiga siku-siku ABC berlaku cosA.cosB = ½. Maka cos(A – B) =…. Bahasan:  siku-siku ABC; cosA.cosB = ½ maka Δ ABC siku-siku di C  C = 90° A + B + C = 180°  A + B = 90° http://meetabied.wordpress.com
  14. 14. A + B + C = 180°  A + B = 90° A = 90° – B  B = 90° – A cos(A – B) = cosA.cosB + sinA.sinB = ½ + sin(90 – B).sin(90-A) = ½ + cosB.cosA = ½ + ½ = 1 Jadi cos(A – B) = 1 http://meetabied.wordpress.com
  15. 15. Rumus jumlah dan selisih dua sudut tan(  +  ) = tan(  -  ) = http://meetabied.wordpress.com
  16. 16. 1. tan 105 ° = …. Bahasan: tan105 ° = tan(60 ° + 45 ° ) http://meetabied.wordpress.com
  17. 17. tan 105 ° = x = = = = -2 - √3 http://meetabied.wordpress.com
  18. 18. 2. Diketahui A + B = 135 ° dan tan B = ½. Nilai tan A= …. Bahasan: A + B = 135° tan(A + B) = tan 135° = -1 = -1 http://meetabied.wordpress.com
  19. 19. = -1 tan A + ½ = -1 + ½ tan A tan A - ½ tan A = -1 - ½ ½ tan p = -1 ½ Jadi, tan p = -3 http://meetabied.wordpress.com
  20. 20. 3. Jika tan q = ½ dan p – q = ¼ π maka tan p = …. Bahasan: p – q = ¼ π tan(p – q) = tan ¼ π = 1 = 1 http://meetabied.wordpress.com
  21. 21. = 1 tan p - ½ = 1 + ½ tan p tan p - ½ tan p = 1 + ½ ½ tan p = 1 ½ Jadi, tan p = 3 http://meetabied.wordpress.com
  22. 22. Rumus Sudut Rangkap sin2a = 2 sina.cosa contoh : 1. sin10 ° = 2sin5 ° .cos5 ° 2. sin6P = 2sin3P.cos3P 3. sin t = 2sin ½ t.cos ½ t http://meetabied.wordpress.com
  23. 23. <ul><li>Diketahui cos  = </li></ul><ul><li>Nilai sin 2  =…. </li></ul><ul><li>Bahasan: </li></ul><ul><li>cos  = </li></ul><ul><li>sin  = </li></ul>4 3 5 http://meetabied.wordpress.com 
  24. 24. cos  = sin  = Jadi sin2  = 2sin  .cos  = 2. x = 4 3 5 http://meetabied.wordpress.com 
  25. 25. 2. Jika tan A = ½ maka sin 2A =…. Bahasan: tan A = ½ sinA = dan cosA = sin2A = 2 sinA.cosA = 2 x x = 1 2 http://meetabied.wordpress.com A
  26. 26. 3. Jika sinx – cosx = p maka harga sin 2x =…. Bahasan: sinx – cosx = p (sinx – cosx) 2 = p 2 sin 2 x – 2sinx.cosx + cos 2 x = p 2 http://meetabied.wordpress.com
  27. 27. sin 2 x – 2sinx.cosx + cos 2 x = p 2 sin 2 x + cos 2 x – 2sinx.cosx = p 2 1 – sin2x = p 2 1 – p 2 = sin2x Jadi, harga sin2x = 1 – p 2 http://meetabied.wordpress.com
  28. 28. 4. Diketahui A adalah sudut lancip dan cos ½ A = Nilai sin A = …. Bahasan: cos ½ A = dengan phytagoras t 2 = 2x – (x + 1) t = √x - 1 √ x+ 1 √ 2x t = √x - 1 http://meetabied.wordpress.com ½A
  29. 29. cos ½ A =  sin ½ A = sinA = 2sin ½ A.cos ½ A = 2 x x = Jadi, sin2x = √ x+ 1 √ 2x t = √x - 1 http://meetabied.wordpress.com ½A
  30. 30. Rumus Sudut Rangkap cos 2a = cos 2 a – sin 2 a = 2cos 2 a – 1 = 1 – 2sin 2 a http://meetabied.wordpress.com
  31. 31. <ul><li>Diketahui cos  = </li></ul><ul><li>maka cos 2  =…. </li></ul><ul><li>Bahasan: </li></ul><ul><li>cos2  = 2cos 2  - 1 </li></ul><ul><li>= 2( ) 2 – 1 </li></ul><ul><li>= - 1 </li></ul><ul><li>= - </li></ul>http://meetabied.wordpress.com
  32. 32. 2. Diketahui sinx = ½ maka cos 2x =…. Bahasan: cos2x = 1 – 2sin 2 x = 1 – 2( ½ ) 2 = 1 – ½ = ½ http://meetabied.wordpress.com
  33. 33. 3. Diketahui tan p = ½ maka cos 2p =…. Bahasan: tan p = ½  cos2p = 1 – 2sin 2 p = 1 – 2( ) 2 = 1 – = 1 2 √ 5 sin p = http://meetabied.wordpress.com p
  34. 34. 4. Diketahui sudut lancip A dengan cos 2A = Nilai tan A = …. Bahasan: • cos 2A = 1 – 2sin 2 A = 1 – 2sin 2 A 2sin 2 A = 1 – = http://meetabied.wordpress.com
  35. 35. • cos 2A = 2cos 2 A – 1 = 2cos 2 A – 1 2cos 2 A = + 1 = tan 2 A = = tan 2 A = ½ A lancip  Jadi , tan A = ½√2 http://meetabied.wordpress.com 2sin 2 A 2cos 2 A
  36. 36. 5. Diketahui A adalah sudut lancip dan cos ½ A = Nilai sin A adalah…. Bahasan: cos A = 2cos 2 ½ A – 1 = 2 - 1 = 2 - 1 = http://meetabied.wordpress.com
  37. 37. cos x = 2 - 1 cos x = cos x =  Jadi, nilai sin x = 1 x √ x 2 – 1 http://meetabied.wordpress.com x
  38. 38. 6. Buktikan: Bahasan: http://meetabied.wordpress.com
  39. 39. Terbukti : http://meetabied.wordpress.com
  40. 40. Rumus Sudut Rangkap tan 2a = Contoh: 1. tan 20 ° = 2. tan 10x = http://meetabied.wordpress.com
  41. 41. 1. Jika tan A = 3 maka tan 2A =…. Bahasan: tan 2A = = = = http://meetabied.wordpress.com
  42. 42. 2. Jika cos x = maka tan 2x =…. Bahasan: tan 2x = = = 5 13 12 http://meetabied.wordpress.com x tan x =
  43. 43. tan 2x = = = Jadi, tan 2x = http://meetabied.wordpress.com
  44. 44. <ul><li>SELAMAT BELAJAR </li></ul>http://meetabied.wordpress.com

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