Please help me with

Question

cos(2Ɵ)+cos(Ɵ)=0.?

 

0 degrees is less than or equal to Ɵ < 360 degrees. Multiple answers are separated by commas.  

 

Answer 1

The trigonometric equation is cos(2θ) + cos(θ) = 0

Recall the double angle identity cos(2x) = 2cos²(x)  - 1

2cos²(θ) - 1  + cos(θ) = 0

2cos²(θ) + cos(θ) - 1 = 0

2cos²(θ) + 2cos(θ) - cos(θ) - 1 = 0

2cos(θ) [cos(θ) + 1] - 1[cos(θ) + 1] = 0

[cos(θ) + 1][2cos(θ) - 1] = 0

Apply zero product property.

[cos(θ) + 1] = 0 and [2cos(θ) - 1] = 0

 

Case 1 : Solve cos(θ) + 1 = 0

cos(θ) = - 1

θ = cos^-1(-1)

θ = cos^-1(cos π)

θ = π

The solutions in the interval 0^o ≤ θ < 360^o is π.

 

Case 2 : Solve 2cos(θ) - 1 = 0

2cos(θ) = 1

cos(θ) = 1/2

cos(θ) = cos (π/3)

The general solution of cos(θ) = cos(α) is θ = 2nπ ± α, where n is an integer.

α = π/3

For n = 0, θ = 2(0)π ± π/3 = ± π/3

For n = 1, θ = 2(1)π ± π/3 = 7π/3 and 5π/3

The solutions in the interval 0^o ≤ θ < 360^o are π/3 and 5π/3.

The solutions in the interval 0^o ≤ θ < 360^o are π, π/3 and 5π/3.