Bonjour,
1) sin(π/3 + 2a) = sin(π/3)cos(2a) + cos(π/3)sin(2a) = √(3)/2 x cos(2a) + 1/2 x sin(2a)
sin(π/3 - 2a) = sin(π/3)cos(2a) - cos(π/3)cos(2a) = √(3)/2 x cos(2a) - 1/2 x sin(2a)
En faisant la différence :
sin(π/3 + 2a) - sin(π/3 - 2a) = 2 x 1/2 x sin(2a) = sin(2a)
de même : cos(π/3 + 2a) + cos(π/3 - 2a)
= [cos(π/3)cos(2a) - sin(π/3)sin(2a)] + [cos(π/3)cos(2a) + sin(π/3)sin(2a)]
= 2cos(π/3)cos(2a)
= 2 x 1/2 x cos(2a)
= cos(2a)
2) on va poser X = 2x
⇒ sin(6x)/sin(2x) - cos(6x)/cos(2x) = sin(3X)/sin(X) - cos(3X)/cos(X)
= sin(2X + X)/sin(X) - cos(2X + X)/cos(X)
= [sin(2X)cos(X) + cos(2X)sin(X)]/sin(X) - [cos(2X)cos(X) - sin(2X)sin(X)]/cos(X)
= sin(2X)cos(X)/sin(X) + cos(2X) - cos(2X) + sin(2X)sin(X)/cos(X)
= sin(2X)[cos(X)/sin(X) + sin(X)/cos(X)]
= sin(2X)[(cos²(X) + sin²(X))/sin(X)cos(X)]
= sin(2X)/(sin(X)cos(X)
= 2sin(X)cos(X)/sin(X)cos(X) (sin(2a) = 2sin(a)cos(a))
= 2
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Bonjour,
1) sin(π/3 + 2a) = sin(π/3)cos(2a) + cos(π/3)sin(2a) = √(3)/2 x cos(2a) + 1/2 x sin(2a)
sin(π/3 - 2a) = sin(π/3)cos(2a) - cos(π/3)cos(2a) = √(3)/2 x cos(2a) - 1/2 x sin(2a)
En faisant la différence :
sin(π/3 + 2a) - sin(π/3 - 2a) = 2 x 1/2 x sin(2a) = sin(2a)
de même : cos(π/3 + 2a) + cos(π/3 - 2a)
= [cos(π/3)cos(2a) - sin(π/3)sin(2a)] + [cos(π/3)cos(2a) + sin(π/3)sin(2a)]
= 2cos(π/3)cos(2a)
= 2 x 1/2 x cos(2a)
= cos(2a)
2) on va poser X = 2x
⇒ sin(6x)/sin(2x) - cos(6x)/cos(2x) = sin(3X)/sin(X) - cos(3X)/cos(X)
= sin(2X + X)/sin(X) - cos(2X + X)/cos(X)
= [sin(2X)cos(X) + cos(2X)sin(X)]/sin(X) - [cos(2X)cos(X) - sin(2X)sin(X)]/cos(X)
= sin(2X)cos(X)/sin(X) + cos(2X) - cos(2X) + sin(2X)sin(X)/cos(X)
= sin(2X)[cos(X)/sin(X) + sin(X)/cos(X)]
= sin(2X)[(cos²(X) + sin²(X))/sin(X)cos(X)]
= sin(2X)/(sin(X)cos(X)
= 2sin(X)cos(X)/sin(X)cos(X) (sin(2a) = 2sin(a)cos(a))
= 2