Questions: Approximate the solutions to the following equation on the interval [0,2 π). Round your answer to four decimal places. sin θ=-0.57 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 as needed.) B. There is no solution.

Approximate the solutions to the following equation on the interval [0,2 π). Round your answer to four decimal places.
sin θ=-0.57
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 as needed.)
B. There is no solution.

Transcript text: Approximate the solutions to the following equation on the interval $[0,2 \pi)$. Round your answer to four decimal places. \[ \boldsymbol{\operatorname { s i n }} \theta=-0.57 \] Select the correct choice and fill in any answer boxes in your choice. A. $\theta \approx$ $\square$ (Type your answer in radians. Round to four decimal places as needed. Use a comma to separate answers as needed.) B. There is no solution.

Solution

Solution Steps

To approximate the solutions to the equation $\sin \theta = -0.57$ on the interval $[0, 2\pi)$, we need to find the angles $\theta$ for which the sine value is $-0.57$. 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.

Step 1: Finding the Principal Value

To solve the equation $ \sin \theta = -0.57 $, we first find the principal value using the inverse sine function. The principal value is given by: \[ \theta_1 = \arcsin(-0.57) \approx 5.6767 \]

Step 2: Finding the Second Solution

Since the sine function is negative in the third and fourth quadrants, we can find the second solution using the property of the sine function: \[ \theta_2 = \pi - \theta_1 \approx 3.7481 \]

Step 3: Adjusting the Principal Value

The principal value $ \theta_1 $ is already in the interval $[0, 2\pi)$. Therefore, we do not need to adjust it further.