To find all exact solutions to the equation csc^2(x) - 3 = -1 on the interval 0 ≤ x < 2, we can follow these steps:
1. Start by isolating the csc^2(x) term by adding 3 to both sides of the equation:
csc^2(x) = -1 + 3
csc^2(x) = 2
2. Next, recall that csc(x) is the reciprocal of sin(x). Therefore, csc^2(x) = 1/sin^2(x).
3. Substitute csc^2(x) = 2 with 1/sin^2(x) = 2:
1/sin^2(x) = 2
1 = 2sin^2(x)
sin^2(x) = 1/2
4. Take the square root of both sides to solve for sin(x):
sin(x) = ±√(1/2)
sin(x) = ±1/√2
sin(x) = ±√2/2
5. Since we are looking for solutions on the interval 0 ≤ x < 2, we need to find the reference angles in the first quadrant where sin(x) is positive.
6. The reference angle where sin(x) is positive and equal to √2/2 is π/4 radians (45 degrees).
7. Therefore, the solutions in the interval 0 ≤ x < 2 are:
x = π/4, 3π/4
These are the exact solutions for the given equation csc^2(x) - 3 = -1 on the interval 0 ≤ x < 2.
