Questions: Solve the equation for exact solutions over the interval [0,2π). 2 cot x + 1 = -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 exact answer, using π as needed. Type your answer in radians. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The solution is the empty set.

$Solve the equation for exact solutions over the interval [0,2π). 2 cot x + 1 = -1 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an exact answer, using π as needed. Type your answer in radians. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The solution is the empty set.$

Transcript text: Solve the equation for exact solutions over the interval $[0,2 \pi)$. \[ 2 \cot x+1=-1 \] 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 exact answer, using $\pi$ as needed. Type your answer in radians. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The solution is the empty set.

Solution

Solution Steps

Step 1: Isolate the Trigonometric Function

We start with the equation: $2 \cot x + 1 = -1$ Rearranging gives: $2 \cot x = -2$ Thus, we have: $\cot x = -1$

Step 2: Convert to Tangent

Using the identity $\cot x = \frac{1}{\tan x}$ , we can rewrite the equation as: $\frac{1}{\tan x} = -1$ This leads to: $\tan x = -1$

Step 3: Find Solutions in the Interval

The solutions to $\tan x = -1$ occur at: $x = \frac{3\pi}{4} \quad \text{and} \quad x = \frac{7\pi}{4}$ These angles are within the interval $[0, 2\pi)$ .