
Question content area top

Part 1

Solve the following equation on the interval 

​[0,2piπ​).

cotangentcot​(33​x)equals=11

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Part 1

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

xequals=nothingenter your response here

​(Type an exact​ answer, using 

piπ 

as needed. Use a comma to separate answers as needed. Type your answer in radians. Simplify your answer. Use integers or fractions for any numbers in the​ expression.)

B.

There is no solution.

To solve the equation \( \cot \left( \frac{3x}{3} \right) = 1 \) on the interval \( [0, 2\pi) \), let's proceed as follows:

1. **Rewrite the Equation**:
   The equation \( \cot \left( \frac{3x}{3} \right) = 1 \) simplifies to \( \cot(3x) = 1 \).

2. **Solve for \( 3x \) Using Cotangent Values**:
   The cotangent function equals 1 at specific angles within a period. We know:
   \[
   \cot \left( \frac{\pi}{4} \right) = 1
   \]
   This means:
   \[
   3x = \frac{\pi}{4} + n\pi \quad \text{for integer } n
   \]

3. **Solve for \( x \)**:
   Divide both sides by 3 to isolate \( x \):
   \[
   x = \frac{\pi}{12} + \frac{n\pi}{3}
   \]

4. **Find All Solutions in \( [0, 2\pi) \)**:
   Now we calculate specific values of \( x \) by substituting integer values of \( n \) to ensure \( x \) is within \( [0, 2\pi) \).

   - For \( n = 0 \): 
     \[
     x = \frac{\pi}{12}
     \]
   - For \( n = 1 \): 
     \[
     x = \frac{\pi}{12} + \frac{\pi}{3} = \frac{5\pi}{12}
     \]
   - For \( n = 2 \): 
     \[
     x = \frac{\pi}{12} + \frac{2\pi}{3} = \frac{9\pi}{12} = \frac{3\pi}{4}
     \]
   - For \( n = 3 \): 
     \[
     x = \frac{\pi}{12} + \pi = \frac{13\pi}{12}
     \]
   - For \( n = 4 \): 
     \[
     x = \frac{\pi}{12} + \frac{4\pi}{3} = \frac{17\pi}{12}
     \]
   - For \( n = 5 \): 
     \[
     x = \frac{\pi}{12} + \frac{5\pi}{3} = \frac{21\pi}{12} = \frac{7\pi}{4}
     \]

   All these values are in the interval \( [0, 2\pi) \).

5. **Answer**:
   Thus, the solutions are:
   \[
   x = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12}, \frac{17\pi}{12}, \frac{7\pi}{4}
   \]

So, the correct choice is **A**: \( x = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12}, \frac{17\pi}{12}, \frac{7\pi}{4} \).

Would you like further details or have any questions?

---

1. What are the general solutions to \( \cot(3x) = 1 \) in all intervals?
2. How does the period of cotangent affect the solutions in \( [0, 2\pi) \)?
3. Why is \( x = \frac{\pi}{12} + \frac{n\pi}{3} \) the correct form for solutions?
4. How can we verify each solution is valid within \( [0, 2\pi) \)?
5. What are other methods to solve trigonometric equations?

**Tip**: Always check the interval of the solution range to ensure all values fit within it.

Solve the equation on the interval [0,2π): cot(3x) = 1. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. x = (Type an exact answer, using π as needed. Use a comma to separate answers as needed. Type your answer in radians. Simplify your answer. Use integers or fractions for any numbers in the expression.)
B. There is no solution.

Learn how to solve the trigonometric equation cot(3x) = 1 within the interval [0, 2π) with a detailed solution. This example covers cotangent properties and finding multiple angle solutions within a specified range. Ideal for students in Grades 11-12.

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