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To solve the equation 4sin(θ) + 5 = 3 on the interval 0 ≤ θ < 2, follow these steps:
1. Subtract 5 from both sides of the equation:
4sin(θ) + 5 - 5 = 3 - 5
4sin(θ) = -2
2. Divide by 4 to isolate sin(θ):
(4sin(θ))/4 = -2/4
sin(θ) = -1/2
3. To find the values of θ on the interval 0 ≤ θ < 2 where sin(θ) = -1/2, recall that sin(θ) = -1/2 in the third and fourth quadrants.
4. In the third quadrant (180° < θ < 270°), sin(θ) = -1/2 when θ = 210°.
5. In the fourth quadrant (270° < θ < 360°), sin(θ) = -1/2 when θ = 330°.
Therefore, the solutions to the equation 4sin(θ) + 5 = 3 on the interval 0 ≤ θ < 2 are θ = 210° and θ = 330°.
