1a) Vérifier que `(5pi)/(12)= (pi)/4 +(pi)/6 `

b) Calculer `cos((5pi)/(12) )` ;`sin((5pi)/(12)) ` ; `tan((5pi)/(12) )`

2a) Vérifier que `(13pi)/(12)= (pi)/3 +(3pi)/4 `

b) Calculer `cos((13pi)/(12) )` , `sin((13pi)/(12) )` , `tan((13pi)/(12)) `

1a) Vérifier que `(5pi)/(12)= (pi)/4 +(pi)/6 `

on a `(pi)/4 +(pi)/6 = (3pi)/(12) + (2pi)/(12)`

` = (3pi +2pi)/(12) = (5pi)/(12) `

alors `=> (5pi)/(12)= (pi)/4 +(pi)/6 `

1 b) Calculer `cos((5pi)/(12) )` ;`sin((5pi)/(12)) ` ; `tan((5pi)/(12) )`

Rappels

1 ` cos(a+b) = cosacosb -sinasinb `

2 ` sin(a+b) = sinacosb + sinbcosa `

3 ` cos((pi)/4) = sin((pi)/4)= (sqrt(2))/2 ` et ` sin((pi)/6)= 1/2 ` , ` cos((pi)/6) = (sqrt(3))/2 `

4 ` tan(a+b)= (tana+tanb)/(1-tanaxxtanb) : a +b ne (pi)/2+kpi ; a ne (pi)/2 +kpi ; b ne (pi)/2 +kpi `


Calculons `cos((5pi)/(12))`

comme ` (5pi)/(12)= (pi)/4 +(pi)/6`

`=> cos((5pi)/(12)) = cos((pi)/4 +(pi)/6)`

` = cos((pi)/4) cos((pi)/6) - sin((pi)/4) sin((pi)/6) `

` = (sqrt(2))/2xx(sqrt(3))/2 - (sqrt(2))/2 xx1/2 `

` = (sqrt(3xx2))/4 - (sqrt(2))/4 `

` = (sqrt(6) -sqrt(2))/4 `

alors `=> cos((5pi)/(12)) = (sqrt(6) -sqrt(2))/4`

Calculons `sin((5pi)/(12))`

on a ` sin((5pi)/(12)) = sin((pi)/4 +(pi)/6)`

` = sin((pi)/4) cos((pi)/6) + sin((pi)/6) cos((pi)/4) `

` = (sqrt(2))/2xx(sqrt(3))/2 + 1/2xx(sqrt(2))/2 `

` = (sqrt(2xx3))/4 + (sqrt(2))/4 `

` = (sqrt(6) + sqrt(2))/4 `

`=> sin((5pi)/(12)) = (sqrt(6) +sqrt(2))/4`

Calculons `tan((5pi)/(12))` : Méthode 1

on a ` tan((5pi)/(12)) = (sin((5pi)/(12)))/(cos((5pi)/(12)))`

` = [ (sqrt(6) + sqrt(2))/4]/[ (sqrt(6) - sqrt(2))/4] `

` = (sqrt(6) +sqrt(2))/(sqrt(6) -sqrt(2)) `

` = (sqrt(6) +sqrt(2))^2/( 6 -2) `

` = ( 8 +2sqrt(12))/4 `

` = (8 + 4sqrt(3))/4 `

` = 2 +sqrt(3) `

alors `=> tan((5pi)/(12)) = 2+sqrt(3)`

Calculons `tan((5pi)/(12))` : Méthode 2

on a `tan((5pi)/(12)) = (tan((pi)/4) + tan((pi)/6))/(1 -tan((pi)/4)tan((pi)/6) `

` = (1 + (sqrt(3))/3) /(1 - (sqrt(3))/3) `

` = [(3 + sqrt(3))/3]/[(3 - sqrt(3))/3]`

` = (3+sqrt(3))/(3-sqrt(3))`

` = (3+sqrt(3))^2/( 9 -3) `

` = (12+6sqrt(3))/6 `

` = 2 + sqrt(3) `

alors `=> tan((5pi)/(12)) = 2+sqrt(3)`

2a) Vérifier que `(13pi)/(12)= (pi)/3 +(3pi)/4 `

on a ` (pi)/3 + (3pi)/4 = (4pi+9pi)/4 = (13pi)/4 `

alors `(13pi)/(12)= (pi)/3 +(3pi)/4 `

2 b) Calculer `cos((13pi)/(12) )` , `sin((13pi)/(12) )` , `tan((13pi)/(12)) `

On a ` cos((3pi)/4) = cos( pi -(pi)/4)= -cos((pi)/4)= -(sqrt(2))/2 `

et ` sin((3pi)/4)= sin(pi -(pi)/4)= sin((pi)/4)= (sqrt(2))/2 `

Calculons : `cos((13pi)/(12) )`

On a `cos((13pi)/(12) ) = cos((pi)/3+(3pi)/4) `

` = cos((pi)/3)cos((3pi)/4) -sin((pi)/3)sin((3pi)/4) `

`= 1/2xx(-sqrt(2))/2 - ( sqrt(3))/2xx(sqrt(2))/2 `

` = (-sqrt(2))/4 - (sqrt(6))/4`

alors `=> cos((13pi)/(12) ) = - (sqrt(6) +sqrt(2))/4 `

Calculons : `sin((13pi)/(12) )`

On a `sin((13pi)/(12) ) = sin((pi)/3+(3pi)/4) `

` = sin((pi)/3)cos((3pi)/4) +cos((pi)/3)sin((3pi)/4) `

`= (sqrt(3))/2xx(-sqrt(2))/2 + 1/2xx(sqrt(2))/2 `

` = (-sqrt(6))/4 + (sqrt(2))/4 `

alors `=> sin((13pi)/(12) ) = (sqrt(2) - sqrt(6))/4 `

Calculons : `tan((13pi)/(12) )`

On a `tan((13pi)/(12) ) = tan((pi)/3+(3pi)/4) `

` = (tan((pi)/3) + tan((pi)/4)) /(1 -tan((pi)/3)tan((pi)/4)) `

` = (sqrt(3) +1)/(1 -sqrt(3))`

` = (1+sqrt(3))^2/(1-3) `

` = (4+2sqrt(3))/(-2) `

` = -2 -sqrt(3) `

alors `=> tan((13pi)/(12)) = -2-sqrt(3) `