Questions: Approximate the solutions to the following equation on the interval [0,2 π). Round your answer to four decimal places. sin θ = -0.41 Select the correct choice and fill in any answer boxes in your choice. A. θ ≈ (Type your answer in radians. Round to four decimal places as needed. Use a comma to separate answers B. There is no solution.

Solution Steps

To solve the equation \(\sin \theta = -0.41\) on the interval \([0, 2\pi)\), we need to find the angles \(\theta\) for which the sine value is \(-0.41\). Since the sine function is periodic and symmetric, there will be two solutions within one period \([0, 2\pi)\). We can use the inverse sine function to find the principal value and then determine the second solution using the properties of the sine function.

We start with the equation \[ \sin \theta = -0.41 \] and we need to find the solutions for \(\theta\) in the interval \([0, 2\pi)\).

Using the inverse sine function, we find the principal value: \[ \theta_1 = \arcsin(-0.41) \approx -0.4260 \] Since this value is negative, we adjust it to fit within the interval \([0, 2\pi)\): \[ \theta_1 = -0.4260 + 2\pi \approx 5.8607 \]

The sine function is symmetric, so the second solution can be found using: \[ \theta_2 = \pi - \theta_1 \approx \pi - 5.8607 \approx 3.564 \]

Final Answer