Solve the following equation for solutions over the interval left bracket 0 comma 2 pi right parenthesis by first solving for the trigonometric function.
2 tangent x plus 3 equals 1
Question content area bottom
Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution set is StartSet nothing EndSet .
  
​(Simplify your answer. Type an exact​ answer, using pi as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as​ needed.)
B.
The solution set is the empty set.


To solve the equation \( 2 \tan(x) + 3 = 1 \) for \( x \) over the interval \( [0, 2\pi) \), we first solve for \( \tan(x) \).

1. Subtract 3 from both sides of the equation:
   \[
   2 \tan(x) + 3 - 3 = 1 - 3
   \]
   \[
   2 \tan(x) = -2
   \]

2. Divide both sides by 2:
   \[
   \tan(x) = -1
   \]

Now we need to find the values of \( x \) in the interval \( [0, 2\pi) \) where \( \tan(x) = -1 \).

The tangent function is negative in the second and fourth quadrants. The general solutions for \( \tan(x) = -1 \) are:
   \[
   x = \frac{3\pi}{4} + k\pi
   \]
where \( k \) is an integer.

We need to find the specific solutions in the interval \( [0, 2\pi) \):

- For \( k = 0 \):
  \[
  x = \frac{3\pi}{4}
  \]

- For \( k = 1 \):
  \[
  x = \frac{3\pi}{4} + \pi = \frac{3\pi}{4} + \frac{4\pi}{4} = \frac{7\pi}{4}
  \]

So, the solutions to the equation in the interval \( [0, 2\pi) \) are \( x = \frac{3\pi}{4} \) and \( x = \frac{7\pi}{4} \).

Therefore, the solution set is:
\[
\left\{ \frac{3\pi}{4}, \frac{7\pi}{4} \right\}
\]

So the correct choice is:
A. The solution set is \( \left\{ \frac{3\pi}{4}, \frac{7\pi}{4} \right\} \).

Would you like more details or have any questions?

Here are 5 numbered questions for you to consider:
1. How do you solve equations involving sine and cosine functions?
2. What are the general solutions for \( \sin(x) = 0 \) and \( \cos(x) = 0 \)?
3. Can you explain how to graph the function \( \tan(x) \)?
4. How do you solve trigonometric equations using identities?
5. What are the properties of the tangent function?

Tip: Always check the interval when finding solutions to trigonometric equations to ensure all solutions are within the given range.

Learn how to solve the trigonometric equation 2 tan(x) + 3 = 1 over the interval [0, 2π). This problem involves finding solutions using the tangent function, with detailed steps provided. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation