Question

Solve the equation for all values of θ, where, 0 ≤ θ≤2 π.
2 sin²θ+cos² θ-1=1/2

Answer 1

The equation 2sin²θ + cos²θ -1 = 1/2 has no solutions for θ in the given range (0 ≤ θ ≤ 2π).

Step-by-step explanation:

To solve the equation 2sin²θ + cos²θ -1 = 1/2, we need to simplify it and solve for θ.

First, we can simplify the equation by combining like terms:

• 2sin²θ + cos²θ -1 - 1/2 = 0
• 2sin²θ + cos²θ - 3/2 = 0

Next, we can use the trigonometric identity sin²θ + cos²θ = 1 to further simplify the equation:

• 2(1 - cos²θ) + cos²θ - 3/2 = 0
• 2 - 2cos²θ + cos²θ - 3/2 = 0
• -2cos²θ - 1/2 = 0
• -4cos²θ - 1 = 0
• -4cos²θ = 1
• cos²θ = -1/4

Since the cosine squared cannot be negative, there are no solutions for θ in the given range (0 ≤ θ ≤ 2π). Therefore, the equation has no solutions.