Questions: Solve the equation for exact solutions over the interval [0°, 360°). 2 sin 2 theta = -1 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or a decimal. Type your answer in degrees. Do not include the degree symbol in your answer. Use a comma to separate answers as needed.) B. The solution is the empty set.

Solve the equation for exact solutions over the interval [0°, 360°).

2 sin 2 theta = -1

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is .
(Type an integer or a decimal. Type your answer in degrees. Do not include the degree symbol in your answer. Use a comma to separate answers as needed.)
B. The solution is the empty set.

Transcript text: Solve the equation for exact solutions over the interval $\left[0^{\circ}, 360^{\circ}\right)$. \[ 2 \sin 2 \theta=-1 \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \{ $\square$ \}. (Type an integer or a decimal. Type your answer in degrees. Do not include the degree symbol in your answer. Use a comma to separate answers as needed.) B. The solution is the empty set.

Solution

Solution Steps

Step 1: Isolate the Sine Function

Starting with the equation: \[ 2 \sin 2\theta = -1 \] we isolate the sine function: \[ \sin 2\theta = -\frac{1}{2} \]

Step 2: Determine Reference Angles

The reference angle for which $\sin x = \frac{1}{2}$ is $30^\circ$. Since the sine function is negative in the third and fourth quadrants, we find the angles: \[ 2\theta = 210^\circ \quad \text{and} \quad 2\theta = 330^\circ \]

Step 3: Solve for $\theta$

Dividing both angles by 2 to solve for $\theta$: \[ \theta_1 = \frac{210^\circ}{2} = 105^\circ \] \[ \theta_2 = \frac{330^\circ}{2} = 165^\circ \]