Bonjour ;
1)
On a :
cos(2a) = cos²(a) - sin²(a) = 2cos²(a) - 1 ;
cos(3a) = cos(2a + a) = cos(2a) cos(a) - sin(2a) sin(a)
= (2cos²(a) - 1) cos(a) - 2sin²(a) cos(a)
= 2cos³(a) - cos(a) - 2(1 - cos²(a)) cos(a)
= 2cos³(a) - cos(a) - 2cos(a) + 2cos³(a)
= 4cos³(a) - 3cos(a) ;
sin(3a) = sin(2a + a) = sin(2a) cos(a) + cos(2a) sin(a)
= 2sin(a) cos²(a) + (2cos²(a) - 1 ) sin(a)
= 2sin(a) cos²(a) + 2sin(a) cos²(a) - sin(a)
= 4sin(a) cos²(a) - sin(a) ;
donc :
cos(5a) = cos(3a + 2a) = cos(3a) cos(2a) - sin(3a) sin(2a)
= (4cos³(a) - 3cos(a)) (2cos²(a) - 1) - 2sin(a) cos(a) (4sin(a) cos²(a) - sin(a))
= 8cos^5(a) - 4cos³(a) - 6cos³(a) + 3cos(a) - 8sin²(a) cos³(a) + 2sin²(a) cos(a)
= 8cos^5(a) - 4cos³(a) - 6cos³(a) + 3cos(a)
- 8(1 - cos²(a)) cos³(a) + 2(1 - cos²(a)) cos(a)
- 8cos³(a) + 8cos^5(a) + 2cos(a) - 2cos³(a)
= 16cos^5(a) - 20cos³(a) + 5cos(a) .
2)
(x + 1) ( 4x² - 2x - 1 )²
= (x + 1) ((4x²)² + (2x)² + (- 1)² + 2(4x²) (- 2x) + 2(4x²) (- 1) + 2(- 2x) (- 1))
= (x + 1) (16x^4 + 4x² + 1 - 16x³ - 8x² + 4x)
= (x + 1) (16x^4 - 16x³ - 4x² + 4x + 1)
= 16x^5 - 16x^4 - 4x³ + 4x² + x + 16x^4 - 16x³ - 4x² + 4x + 1
= 16x^5 - 20x³ + 5x + 1 .
3)
- 1 = cos(π) = cos(5 * π/5) = 16cos^5(π/5) - 20cos³(π/5) + 5cos(π/5) ;
16cos^5(π/5) - 20cos³(π/5) + 5cos(π/5) + 1 = 0 ;
(cos(π/5) + 1) (4cos²(π/5) - 2cos(π/5) - 1) = 0 ;
4cos²(π/5) - 2cos(π/5) - 1 = 0 car cos(π/5) + 1 ≠ 0 ;
donc t = cos(π/5) est solution de l'équation en x : 4x² - 2x - 1 = 0 .
On a : 4x² - 2x - 1 = 0 ;
donc : Δ = (- 2)² - 4 * 4 * (- 1) = 4 + 16 = 20 = 4 * 5 = 2² * 5 ;
donc : √Δ = √(2² * 5) = 2√5 ;
donc les solutions de l'équation sont :
x1 = (2 - 2√5)/8 = (1 - √5)/4 < 0 et x2 = (2 + 2√5)/8 = (1 + √5)/4 > 0 .
Comme on a : 0 < π/5 < π/2 alors cos(π/5) > 0 ;
donc : t = cos(π/5) = (1 + √5)/4 .
4)
On a : cos²(π/5) = ((1 + √5)/4)² = (1 + (√5)² + 2√5)/16
= (1 + 5 + 2√5)/16 = (6 + 2√5)/16 = (3 + √5)/8 ;
donc : sin²(π/5) = 1 - cos²(π/5) = 1 - (3 + √5)/8
= (8 - 3 - √5)/8 = (5 - √5)/8 .
Comme on a : 0 < π/5 < π/2 alors sin(π/5) > 0 ;
donc : sin(π/5) = √((5 - √5)/8) .
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Bonjour ;
1)
On a :
cos(2a) = cos²(a) - sin²(a) = 2cos²(a) - 1 ;
cos(3a) = cos(2a + a) = cos(2a) cos(a) - sin(2a) sin(a)
= (2cos²(a) - 1) cos(a) - 2sin²(a) cos(a)
= 2cos³(a) - cos(a) - 2(1 - cos²(a)) cos(a)
= 2cos³(a) - cos(a) - 2cos(a) + 2cos³(a)
= 4cos³(a) - 3cos(a) ;
sin(3a) = sin(2a + a) = sin(2a) cos(a) + cos(2a) sin(a)
= 2sin(a) cos²(a) + (2cos²(a) - 1 ) sin(a)
= 2sin(a) cos²(a) + 2sin(a) cos²(a) - sin(a)
= 4sin(a) cos²(a) - sin(a) ;
donc :
cos(5a) = cos(3a + 2a) = cos(3a) cos(2a) - sin(3a) sin(2a)
= (4cos³(a) - 3cos(a)) (2cos²(a) - 1) - 2sin(a) cos(a) (4sin(a) cos²(a) - sin(a))
= 8cos^5(a) - 4cos³(a) - 6cos³(a) + 3cos(a) - 8sin²(a) cos³(a) + 2sin²(a) cos(a)
= 8cos^5(a) - 4cos³(a) - 6cos³(a) + 3cos(a)
- 8(1 - cos²(a)) cos³(a) + 2(1 - cos²(a)) cos(a)
= 8cos^5(a) - 4cos³(a) - 6cos³(a) + 3cos(a)
- 8cos³(a) + 8cos^5(a) + 2cos(a) - 2cos³(a)
= 16cos^5(a) - 20cos³(a) + 5cos(a) .
2)
On a :
(x + 1) ( 4x² - 2x - 1 )²
= (x + 1) ((4x²)² + (2x)² + (- 1)² + 2(4x²) (- 2x) + 2(4x²) (- 1) + 2(- 2x) (- 1))
= (x + 1) (16x^4 + 4x² + 1 - 16x³ - 8x² + 4x)
= (x + 1) (16x^4 - 16x³ - 4x² + 4x + 1)
= 16x^5 - 16x^4 - 4x³ + 4x² + x + 16x^4 - 16x³ - 4x² + 4x + 1
= 16x^5 - 20x³ + 5x + 1 .
3)
On a :
- 1 = cos(π) = cos(5 * π/5) = 16cos^5(π/5) - 20cos³(π/5) + 5cos(π/5) ;
donc :
16cos^5(π/5) - 20cos³(π/5) + 5cos(π/5) + 1 = 0 ;
donc :
(cos(π/5) + 1) (4cos²(π/5) - 2cos(π/5) - 1) = 0 ;
donc :
4cos²(π/5) - 2cos(π/5) - 1 = 0 car cos(π/5) + 1 ≠ 0 ;
donc t = cos(π/5) est solution de l'équation en x : 4x² - 2x - 1 = 0 .
On a : 4x² - 2x - 1 = 0 ;
donc : Δ = (- 2)² - 4 * 4 * (- 1) = 4 + 16 = 20 = 4 * 5 = 2² * 5 ;
donc : √Δ = √(2² * 5) = 2√5 ;
donc les solutions de l'équation sont :
x1 = (2 - 2√5)/8 = (1 - √5)/4 < 0 et x2 = (2 + 2√5)/8 = (1 + √5)/4 > 0 .
Comme on a : 0 < π/5 < π/2 alors cos(π/5) > 0 ;
donc : t = cos(π/5) = (1 + √5)/4 .
4)
On a : cos²(π/5) = ((1 + √5)/4)² = (1 + (√5)² + 2√5)/16
= (1 + 5 + 2√5)/16 = (6 + 2√5)/16 = (3 + √5)/8 ;
donc : sin²(π/5) = 1 - cos²(π/5) = 1 - (3 + √5)/8
= (8 - 3 - √5)/8 = (5 - √5)/8 .
Comme on a : 0 < π/5 < π/2 alors sin(π/5) > 0 ;
donc : sin(π/5) = √((5 - √5)/8) .