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TRIGONOMETRI

TRIGONOMETRI. Setelah menyaksikan tayangan ini anda dapat Menyelesaikan soal-soal yang berkaitan dengan jumlah dan selisih sudut serta sudut rangkap. Rumus jumlah dan selisih dua sudut sin(  + ) = sin.cos + cos.sin sin( - ) = sin.cos - cos.sin.

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TRIGONOMETRI

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  1. TRIGONOMETRI

  2. Setelah menyaksikan tayangan ini anda dapat Menyelesaikan soal-soal yang berkaitan dengan jumlah dan selisih sudut serta sudut rangkap

  3. Rumus jumlah dan selisih dua sudut sin( + ) = sin.cos + cos.sin sin( - ) = sin.cos - cos.sin

  4. 1. Sin 75o = …. Bahasan: sin( + ) = sin.cos + cos.sin sin750 = sin(450 + 300) = sin450cos300 + cos450sin300 = ½√2.½√3 + ½√2.½ = ¼√6 + ¼√2 = ¼√2(√2 + 1)

  5. A B 2. Diketahui sin A = cos B = A dan B adalah sudut-sudut lancip sin(A – B) =…. Bahasan: sin(A – B)= sinAcosB – cosAsinB sinA = cosA = ? ? cos B = sin B = 5 3 24 25 4 7

  6. sin A =  cos A = cos B =  sin B = sin(A – B) =…. = sinAcosB – cosAsinB = x - x = =

  7. Rumus jumlah dan selisih dua sudut cos( + ) = coscos - sinsin cos( - ) = coscos + sinsin

  8. 1. Bahasan: coscos + sinsin = cos( - ) = = =

  9. 2. a. –sina.sinb b. cosa.cosb c. sina.sinb d. 1 – tana.tanb e. 1 + tana.tanb

  10. = = 1 – tana.tanb  jawab d

  11. 3. Tentukan nilai cos56° + sin56°.tan28° Bahasan: cos56° + sin56°.tan28° = cos56° + sin56°. = cos56° +

  12. = cos56° + = = = Jadi, Nilai cos56° + sin56°.tan28° = 1 = 1

  13. 4. Pada suatu segitiga siku-siku ABC berlaku cosA.cosB = ½. Maka cos(A – B) =…. Bahasan:  siku-siku ABC; cosA.cosB = ½ makaΔABC siku-siku di C C = 90° A + B + C = 180°  A + B = 90°

  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

  15. Rumus jumlah dan selisih dua sudut tan( + ) = tan( - ) =

  16. 1. tan 105° = …. Bahasan: tan105° = tan(60° + 45°)

  17. tan 105° = x = = = = -2 - √3

  18. 2. Diketahui A + B = 135° dan tan B =½. Nilai tan A= …. Bahasan: A + B = 135° tan(A + B) = tan 135° = -1 = -1

  19. = -1 tan A + ½= -1 + ½tan A tan A - ½tan A = -1 - ½ ½tan p = -1½ Jadi, tan p = -3

  20. 3. Jika tan q =½ dan p – q = ¼π maka tan p = …. Bahasan: p – q = ¼π tan(p – q) = tan ¼π = 1 = 1

  21. = 1 tan p - ½ = 1 + ½tan p tan p - ½tan p = 1 + ½ ½tan p = 1½ Jadi, tan p = 3

  22. Rumus Sudut Rangkap sin2a = 2 sina.cosa contoh: 1. sin10° = 2sin5°.cos5° 2. sin6P = 2sin3P.cos3P 3. sin t = 2sin½t.cos½t

  23. • Diketahui cos = • Nilai sin 2 =…. • Bahasan: • cos = • sin = 5 4 3

  24. cos = sin = Jadi sin2 = 2sin.cos = 2. x =  5 4 3

  25. A 2. Jika tan A = ½ maka sin 2A =…. Bahasan: tan A = ½ sinA = dancosA = sin2A = 2 sinA.cosA = 2x x= 1 2

  26. 3. Jika sinx – cosx = p maka harga sin 2x =…. Bahasan: sinx – cosx = p (sinx – cosx)2 = p2 sin2x – 2sinx.cosx + cos2x = p2

  27. sin2x – 2sinx.cosx + cos2x = p2 sin2x + cos2x – 2sinx.cosx = p2 1 – sin2x = p2 1 – p2 = sin2x Jadi, harga sin2x = 1 – p2

  28. ½A 4. Diketahui A adalah sudut lancip dan cos½A = Nilai sin A = …. Bahasan: cos½A = dengan phytagoras t2 = 2x – (x + 1) t = √x - 1 √2x t = √x - 1 √x+ 1

  29. ½A √2x t = √x - 1 √x+ 1 cos½A =  sin½A = sinA = 2sin½A.cos½A = 2 x x = Jadi, sin2x =

  30. Rumus Sudut Rangkap cos 2a = cos2a – sin2a = 2cos2a – 1 = 1 – 2sin2a

  31. Diketahui cos = • maka cos 2 =…. • Bahasan: • cos2 = 2cos2 - 1 • = 2( )2 – 1 • = - 1 • = -

  32. 2. Diketahui sinx = ½ maka cos 2x =…. Bahasan: cos2x = 1 – 2sin2x = 1 – 2(½)2 = 1 – ½ = ½

  33. p 3. Diketahui tan p = ½ maka cos 2p =…. Bahasan: tan p = ½  cos2p = 1 – 2sin2p = 1 – 2( )2 = 1 – = sin p = √5 1 2

  34. 4. Diketahui sudut lancip A dengan cos 2A = Nilai tan A = …. Bahasan: • cos 2A = 1 – 2sin2A = 1 – 2sin2A 2sin2A = 1 – =

  35. 2sin2A 2cos2A • cos 2A = 2cos2A – 1 = 2cos2A – 1 2cos2A = + 1 = tan2A = = tan2A = ½ A lancip  Jadi, tan A = ½√2

  36. 5. Diketahui A adalah sudut lancip dan cos½A = Nilai sin A adalah…. Bahasan: cos A =2cos2½A – 1 = 2 - 1 = 2 - 1 =

  37. x cos x = 2 - 1 cos x = cos x =  Jadi, nilai sin x = x √x2 – 1 1

  38. 6. Buktikan: Bahasan:

  39. Terbukti :

  40. Rumus Sudut Rangkap tan 2a = Contoh: 1. tan 20° = 2. tan 10x =

  41. 1. Jika tan A = 3 maka tan 2A =…. Bahasan: tan 2A = = = =

  42. x tan x = 2. Jika cos x = maka tan 2x =…. Bahasan: tan 2x = = = 13 12 5

  43. tan 2x = = = Jadi, tan 2x =

  44. SELAMAT BELAJAR

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