Question

cos 3 theta + 8 cos ^3 =0

Answer 1

Answer:

♦ Principal solution :

∅ = π/3 , π/2 , 2π/3 , 4π/3 , 3π/2 , 5π/3

♦ General solution :

∅ = 2nπ ± π/2 or 2mπ ± π/3 or 2qπ ± 2π/3

m , n , q € Z

Note:

★ If sin∅ = sinα , then ;

∅ = nπ + (-1)ⁿα , n € Z

★ If cos∅ = cosα , then ;

∅ = 2nπ ± α , n € Z

★ If tan∅ = tanα , then ;

∅ = nπ + α , n € Z

★ sin3∅ = 3sin∅ - 4sin³∅

★ cos3∅ = 4cos³∅ - 3cos∅

Solution:

• Given : cos3∅ + 8cos³∅ = 0
• To find : ∅ = ?

We have ;

=> cos3∅ + 8cos³∅ = 0

=> 4cos³∅ - 3cos∅ + 8cos³∅ = 0

=> 12cos³∅ - 3cos∅ = 0

=> 3cos∅•(4cos²∅ - 1) = 0

=> cos∅•(4cos²∅ - 1) = 0

=> cos∅•(2cos∅ - 1)•(2cos∅ + 1) = 0

Here,

Three cases arises :-

1) cos∅ = 0

OR

2) 2cos∅ - 1 = 0

OR

3) 2cos∅ + 1 = 0

★ Case(1) : cos∅ = 0

• Principal solution :-

=> cos∅ = 0

=> cos∅ = cosπ/2 or cos(2π - π/2)

=> ∅ = π/2 or 3π/2

• General solution :-

=> cos∅ = 0

=> cos∅ = cosπ/2

=> ∅ = 2nπ ± π/2 , n € Z

★ Case(2) : 2cos∅ - 1 = 0

• Principal solution :-

=> 2cos∅ - 1 = 0

=> 2cos∅ = 1

=> cos∅ = 1/2

=> cos∅ = cosπ/3 or cos(2π - π/3)

=> cos∅ = cosπ/3 or cos5π/3

=> ∅ = π/3 or 5π/3

• General solution :-

=> 2cos∅ - 1 = 0

=> 2cos∅ = 1

=> cos∅ = 1/2

=> cos∅ = cosπ/3

=> ∅ = 2mπ ± π/3 , m € Z

★ Case(3) : 2cos∅ + 1 = 0

• Principal solution :-

=> 2cos∅ + 1 = 0

=> 2cos∅ = -1

=> cos∅ = -1/2

=> cos∅ = cos(π - π/3) or cos(π + π/3)

=> cos∅ = cos2π/3 or cos4π/3

=> ∅ = 2π/3 or 4π/3

• General solution :-

=> 2cos∅ + 1 = 0

=> 2cos∅ = -1

=> cos∅ = -1/2

=> cos∅ = cos(π - π/3)

=> cos∅ = cos2π/3

=> ∅ = 2qπ ± 2π/3 , q € Z