Answer:
The solutions to the given equations are:
For equation (a): θ = arcsin(√(1/2)) or θ = π - arcsin(√(1/2))
For equation (b): θ = arccos(√(1/2)) or θ = -arccos(√(1/2))
Explanation:
To solve equation (a), we can rewrite it using the Pythagorean identity:
4(1 - sin^2θ) - 4cosθ + 1 = 0
Expanding and rearranging, we get:
4 - 4sin^2θ - 4cosθ + 1 = 0
-4sin^2θ - 4cosθ + 5 = 0
Now, let's solve equation (b):
4sin^2θ + 2cos^2θ = 3
Using the Pythagorean identity, we can rewrite it as:
4(1 - cos^2θ) + 2cos^2θ = 3
Expanding and rearranging, we get:
4 - 4cos^2θ + 2cos^2θ = 3
-2cos^2θ + 4 = 3
-2cos^2θ = -1
Dividing by -2, we have:
cos^2θ = 1/2
Taking the square root of both sides, we get:
cosθ = ±√(1/2)
Therefore, the solutions to the equations are:
For equation (a): θ = arcsin(√(1/2)) or θ = π - arcsin(√(1/2))
For equation (b): θ = arccos(√(1/2)) or θ = -arccos(√(1/2))