Question

Solve 2cos²(θ)+cos(θ)=0.

a) θ= π/3
b) θ= π/2
c) θ= π
d) θ= 2π/3

Answer 1

To solve the equation 2cos²(θ) + cos(θ) = 0, we can factor out the common term cos(θ) and solve two separate equations. Therefore, the correct answers are θ = π/2 (option b) ) and θ = 2π/3 (option d).

Step-by-step explanation:

To solve the equation 2cos²(θ) + cos(θ) = 0, we can factor out the common term cos(θ):

cos(θ)(2cos(θ) + 1) = 0

This equation will be true if either cos(θ) = 0 or 2cos(θ) + 1 = 0. Solving these two equations separately:

1. If cos(θ) = 0, then θ = π/2. This corresponds to option b) in the given choices.

2. If 2cos(θ) + 1 = 0, then 2cos(θ) = -1, and dividing both sides by 2:

cos(θ) = -1/2

This occurs when θ = 2π/3 or θ = 4π/3. Both of these angles are equivalent mod 2π, so θ = 2π/3 is the correct answer. This corresponds to option d) in the given choices.

Therefore, the correct answers are θ = π/2 (option b) ) and θ = 2π/3 (option d).