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Tentukan nilai x yang memenuhi persamaan berikut untuk 0 ∘ ≤ x ≤ 36 0 ∘ . a. ( sin 3 x − cos 4 x ) ( cos 5 x − sin 4 x ) = 0

Tentukan nilai  yang memenuhi persamaan berikut untuk .

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S. Nur

Master Teacher

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nilai x yang memenuhi persamaan ( sin 3 x − cos 4 x ) ( cos 5 x − sin 4 x ) = 0 untuk 0 ∘ ≤ x ≤ 36 0 ∘ adalah { 12 , 9 ∘ ; 3 0 ∘ ; 6 4 ∘ ; 7 0 ∘ ; 11 0 ∘ ; 115 , 5 ; 15 0 ∘ ; 166 , 6 ∘ ; 19 0 ∘ ; 218 , 1 ∘ ; 23 0 ∘ ; 269 , 2 ∘ ; 27 0 ∘ ; 31 0 ∘ ; 320 , 7 ∘ } .

 nilai  yang memenuhi persamaan untuk  adalah .

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Ingat kembali: cos ( 9 0 ∘ − A ) = sin A cos A − cos B = − 2 ⋅ sin 2 1 ​ ( A + B ) ⋅ sin 2 1 ​ ( A − B ) sin x = sin α { x = α + k ⋅ 36 0 ∘ x = ( 18 0 ∘ − α ) + k ⋅ 36 0 ∘ ​ Diberikan persamaan ( sin 3 x − cos 4 x ) ( cos 5 x − sin 4 x ) = 0 untuk 0 ∘ ≤ x ≤ 36 0 ∘ . Maka: - Untuk ( sin 3 x − cos 4 x ) = 0 ( sin 3 x − cos 4 x ) sin 3 x − cos 4 x cos ( 9 0 ∘ − 3 x ) − cos 4 x − 2 ⋅ sin 2 1 ​ ( 9 0 ∘ + x ) ⋅ sin 2 1 ​ ( 9 0 ∘ − 7 x ) sin 2 1 ​ ( 9 0 ∘ + x ) ⋅ sin 2 1 ​ ( 9 0 ∘ − 7 x ) ​ = = = = = ​ 0 0 0 0 0 ​ Bentuk sin 2 1 ​ ( 9 0 ∘ + x ) = 0 sin 2 1 ​ ( 9 0 ∘ + x ) = 0 sin 2 1 ​ ( 9 0 ∘ + x ) = sin 0 ∘ Persamaan sinus (1) 2 1 ​ ( 9 0 ∘ + x ) 9 0 ∘ + x x ​ = = = ​ 0 ∘ + k ⋅ 36 0 ∘ 0 ∘ + k ⋅ 72 0 ∘ − 9 0 ∘ + k ⋅ 72 0 ∘ ( tidak memenuhi ) ​ Persamaan sinus (2) 2 1 ​ ( 9 0 ∘ + x ) 2 1 ​ ( 9 0 ∘ + x ) 9 0 ∘ + x x ​ = = = = ​ ( 18 0 ∘ − 0 ∘ ) + k ⋅ 36 0 ∘ 18 0 ∘ + k ⋅ 36 0 ∘ 36 0 ∘ + k ⋅ 72 0 ∘ 27 0 ∘ + k ⋅ 72 0 ∘ ​ Bentuk sin 2 1 ​ ( 9 0 ∘ − 7 x ) = 0 sin 2 1 ​ ( 9 0 ∘ − 7 x ) = 0 sin 2 1 ​ ( 9 0 ∘ − 7 x ) = sin 0 ∘ Persamaan sinus (1) 2 1 ​ ( 9 0 ∘ − 7 x ) 9 0 ∘ − 7 x − 7 x x ​ = = = = ​ 0 ∘ + k ⋅ 36 0 ∘ 0 ∘ + k ⋅ 72 0 ∘ − 9 0 ∘ + k ⋅ 72 0 ∘ 12 , 9 − k ⋅ 102 , 6 ∘ ​ Persamaan sinus (2) 2 1 ​ ( 9 0 ∘ − 7 x ) 2 1 ​ ( 9 0 ∘ − 7 x ) 9 0 ∘ − 7 x − 7 x x ​ = = = = = ​ ( 18 0 ∘ − 0 ∘ ) + k ⋅ 36 0 ∘ 180 0 ∘ + k ⋅ 36 0 ∘ 36 0 ∘ + k ⋅ 72 0 ∘ 27 0 ∘ + k ⋅ 72 0 ∘ − 38 , 6 − k ⋅ 102 , 6 ∘ ​ - Untuk ( cos 5 x − sin 4 x ) = 0 ( cos 5 x − sin 4 x ) cos 5 x − sin 4 x cos 5 x − cos ( 9 0 ∘ − 4 x ) − 2 ⋅ sin 2 1 ​ ( x + 9 0 ∘ ) ⋅ sin 2 1 ​ ( 9 x + 9 0 ∘ ) sin 2 1 ​ ( x + 9 0 ∘ ) ⋅ sin 2 1 ​ ( 9 x + 9 0 ∘ ) ​ = = = = = ​ 0 0 0 0 0 ​ Bentuk sin 2 1 ​ ( x + 9 0 ∘ ) = 0 sin 2 1 ​ ( x + 9 0 ∘ ) = 0 sin 2 1 ​ ( x + 9 0 ∘ ) = sin 0 ∘ Persamaan sinus (1) 2 1 ​ ( x + 9 0 ∘ ) x + 9 0 ∘ x ​ = = = ​ 0 ∘ + k ⋅ 36 0 ∘ 0 ∘ + k ⋅ 72 0 ∘ − 9 0 ∘ + k ⋅ 720 ( tidak memenuhi ) ​ Persamaan sinus (2) 2 1 ​ ( x + 9 0 ∘ ) 2 1 ​ ( x + 9 0 ∘ ) x + 9 0 ∘ x ​ = = = = ​ ( 18 0 ∘ − 0 ∘ ) + k ⋅ 36 0 ∘ 18 0 ∘ + k ⋅ 36 0 ∘ 36 0 ∘ + k ⋅ 72 0 ∘ 27 0 ∘ + k ⋅ 36 0 ∘ ​ Bentuk sin 2 1 ​ ( 9 x + 9 0 ∘ ) = 0 sin 2 1 ​ ( 9 x + 9 0 ∘ ) = 0 sin 2 1 ​ ( 9 x + 9 0 ∘ ) = sin 0 ∘ Persamaan sinus (1) 2 1 ​ ( 9 x + 9 0 ∘ ) 9 x + 9 0 ∘ 9 x x ​ = = = = ​ 0 ∘ + k ⋅ 36 0 ∘ 0 ∘ + k ⋅ 72 0 ∘ − 9 0 ∘ + k ⋅ 72 0 ∘ − 1 0 ∘ + k ⋅ 8 0 ∘ ​ Persamaan sinus (2) 2 1 ​ ( 9 x + 9 0 ∘ ) 2 1 ​ ( 9 x + 9 0 ∘ ) 9 x + 9 0 ∘ 9 x x ​ = = = = = ​ ( 18 0 ∘ − 0 ∘ ) + k ⋅ 36 0 ∘ 18 0 ∘ + k ⋅ 36 0 ∘ 36 0 ∘ + k ⋅ 72 0 ∘ 27 0 ∘ + k ⋅ 72 0 ∘ 3 0 ∘ + k ⋅ 8 0 ∘ ​ Jadi,nilai x yang memenuhi persamaan ( sin 3 x − cos 4 x ) ( cos 5 x − sin 4 x ) = 0 untuk 0 ∘ ≤ x ≤ 36 0 ∘ adalah { 12 , 9 ∘ ; 3 0 ∘ ; 6 4 ∘ ; 7 0 ∘ ; 11 0 ∘ ; 115 , 5 ; 15 0 ∘ ; 166 , 6 ∘ ; 19 0 ∘ ; 218 , 1 ∘ ; 23 0 ∘ ; 269 , 2 ∘ ; 27 0 ∘ ; 31 0 ∘ ; 320 , 7 ∘ } .

Ingat kembali:

  •  
  •   
  •  

Diberikan persamaan  untuk . Maka:

- Untuk   

 

Bentuk  

  

  • Persamaan sinus (1)

 

  • Persamaan sinus (2)

 

bold italic k bold equals bold 0 bold rightwards arrow bold italic x bold equals bold 270 bold degree bold plus bold 0 bold times bold 720 bold degree bold equals bold 270 bold degree bold plus bold 0 bold degree bold equals bold 270 bold degree 

Bentuk  

 

  • Persamaan sinus (1)

 

begin mathsize 12px style bold italic k bold equals bold 0 bold rightwards arrow bold italic x bold equals bold 12 bold comma bold 9 bold minus bold 0 bold times bold 102 bold comma bold 6 bold degree bold equals bold 12 bold comma bold 9 bold degree bold minus bold 0 bold degree bold equals bold 12 bold comma bold 9 bold degree bold italic k bold equals bold minus bold 1 bold rightwards arrow bold italic x bold equals bold 12 bold comma bold 9 bold minus bold left parenthesis bold minus bold 1 bold right parenthesis bold times bold 102 bold comma bold 6 bold degree bold equals bold 12 bold comma bold 9 bold degree bold plus bold 102 bold comma bold 6 bold degree bold equals bold 115 bold comma bold 5 bold degree bold italic k bold equals bold minus bold 2 bold rightwards arrow bold italic x bold equals bold 12 bold comma bold 9 bold minus bold left parenthesis bold minus bold 2 bold right parenthesis bold times bold 102 bold comma bold 6 bold degree bold equals bold 12 bold comma bold 9 bold degree bold plus bold 205 bold comma bold 2 bold degree bold equals bold 218 bold comma bold 1 bold degree bold italic k bold equals bold minus bold 3 bold rightwards arrow bold italic x bold equals bold 12 bold comma bold 9 bold minus bold left parenthesis bold minus bold 3 bold right parenthesis bold times bold 102 bold comma bold 6 bold degree bold equals bold 12 bold comma bold 9 bold degree bold plus bold 307 bold comma bold 8 bold degree bold equals bold 320 bold comma bold 7 bold degree end style 

  • Persamaan sinus (2)

 

begin mathsize 12px style bold italic k bold equals bold minus bold 1 bold rightwards arrow bold italic x bold equals bold minus bold 38 bold comma bold 6 bold minus bold left parenthesis bold minus bold 1 bold right parenthesis bold times bold 102 bold comma bold 6 bold degree bold equals bold minus bold 38 bold comma bold 6 bold plus bold 102 bold comma bold 6 bold degree bold equals bold 64 bold degree bold italic k bold equals bold minus bold 2 bold rightwards arrow bold italic x bold equals bold minus bold 38 bold comma bold 6 bold minus bold left parenthesis bold minus bold 2 bold right parenthesis bold times bold 102 bold comma bold 6 bold degree bold equals bold minus bold 38 bold comma bold 6 bold plus bold 205 bold comma bold 2 bold degree bold equals bold 166 bold comma bold 6 bold degree bold italic k bold equals bold minus bold 3 bold rightwards arrow bold italic x bold equals bold minus bold 38 bold comma bold 6 bold minus bold left parenthesis bold minus bold 3 bold right parenthesis bold times bold 102 bold comma bold 6 bold degree bold equals bold minus bold 38 bold comma bold 6 bold plus bold 307 bold comma bold 8 bold degree bold equals bold 269 bold comma bold 2 bold degree end style 

- Untuk  

 

Bentuk 

 

  • Persamaan sinus (1)

  

  • Persamaan sinus (2)

 

bold italic k bold equals bold 0 bold rightwards arrow table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold italic x end table table attributes columnalign right center left columnspacing 0px end attributes row blank bold equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold 270 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold italic k end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold 360 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank bold equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold 270 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold 0 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank bold equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold 270 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold degree end table 

Bentuk  

 

  • Persamaan sinus (1)

 

begin mathsize 12px style bold italic k bold equals bold 1 bold rightwards arrow bold italic x bold equals bold minus bold 10 bold degree bold plus bold 1 bold times bold 80 bold degree bold equals bold minus bold 10 bold degree bold plus bold 80 bold degree bold equals bold 70 bold degree bold italic k bold equals bold 2 bold rightwards arrow bold italic x bold equals bold minus bold 10 bold degree bold plus bold 2 bold times bold 80 bold degree bold equals bold minus bold 10 bold degree bold plus bold 160 bold degree bold equals bold 150 bold degree bold italic k bold equals bold 3 bold rightwards arrow bold italic x bold equals bold minus bold 10 bold degree bold plus bold 3 bold times bold 80 bold degree bold equals bold minus bold 10 bold degree bold plus bold 240 bold degree bold equals bold 230 bold degree bold italic k bold equals bold 4 bold rightwards arrow bold italic x bold equals bold minus bold 10 bold degree bold plus bold 4 bold times bold 80 bold degree bold equals bold minus bold 10 bold degree bold plus bold 320 bold degree bold equals bold 310 bold degree end style 

  • Persamaan sinus (2)

 

begin mathsize 12px style bold italic k bold equals bold 0 bold rightwards arrow bold italic x bold equals bold 30 bold degree bold plus bold 0 bold times bold 80 bold degree bold equals bold 30 bold degree bold plus bold 0 bold equals bold 30 bold degree bold italic k bold equals bold 1 bold rightwards arrow bold italic x bold equals bold 30 bold degree bold plus bold 1 bold times bold 80 bold degree bold equals bold 30 bold degree bold plus bold 80 bold degree bold equals bold 110 bold degree bold italic k bold equals bold 2 bold rightwards arrow bold italic x bold equals bold 30 bold degree bold plus bold 2 bold times bold 80 bold degree bold equals bold 30 bold degree bold plus bold 160 bold degree bold equals bold 190 bold degree bold italic k bold equals bold 3 bold rightwards arrow bold italic x bold equals bold 30 bold degree bold plus bold 3 bold times bold 80 bold degree bold equals bold 30 bold degree bold plus bold 240 bold degree bold equals bold 270 bold degree end style 

Jadi, nilai  yang memenuhi persamaan untuk  adalah .

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