Solve the equation. Give a general formula for all the solutions. List six solutions.

is an integer
Identify the general formula for all the solutions to based on the smaller angle.
(Simplify your answer. Use angle measures greater than or equal to 0 and less than . Type an exact answer, using 1 . as nes needed. Use integers or fractions for any numbers in the expression. Type
an expression using k as the variable.)

Ask by Daniels Moran. in the United States

Jan 24,2025

Question

Solve the equation. Give a general formula for all the solutions. List six solutions.

is an integer
Identify the general formula for all the solutions to based on the smaller angle.
(Simplify your answer. Use angle measures greater than or equal to 0 and less than . Type an exact answer, using 1 . as nes needed. Use integers or fractions for any numbers in the expression. Type
an expression using k as the variable.)

Ask by Daniels Moran. in the United States

Jan 24,2025

UpStudy AI · Accepted Answer

To solve the equation $$ \sin \theta = \frac{1}{2} $$, we'll find all angles $$ \theta $$ that satisfy this condition.

### Step 1: Identify the Reference Angle
The reference angle where $$ \sin \theta = \frac{1}{2} $$ is:
$$ \theta = \frac{\pi}{6} $$

### Step 2: Determine All Solutions in the Interval $$[0, 2\pi)$$
The sine function is positive in the first and second quadrants. Therefore, the solutions within one full rotation ($$0 \leq \theta < 2\pi$$) are:
$$ \theta = \frac{\pi}{6} \quad \text{and} \quad \theta = \frac{5\pi}{6} $$

### Step 3: Generalize for All Solutions
The sine function has a period of $$2\pi$$, meaning the solutions repeat every $$2\pi$$ radians. Therefore, the general formulas for all solutions are:
$$ \theta = \frac{\pi}{6} + 2k\pi $$
$$ \theta = \frac{5\pi}{6} + 2k\pi $$
where $$ k $$ is any integer.

### Step 4: List Six Specific Solutions
By substituting $$ k = 0, 1, 2 $$ into the general formulas, we obtain six specific solutions:

1. For $$ k = 0 $$:
   $$ \theta = \frac{\pi}{6} $$
   $$ \theta = \frac{5\pi}{6} $$

2. For $$ k = 1 $$:
   $$ \theta = \frac{\pi}{6} + 2\pi = \frac{13\pi}{6} $$
   $$ \theta = \frac{5\pi}{6} + 2\pi = \frac{17\pi}{6} $$

3. For $$ k = 2 $$:
   $$ \theta = \frac{\pi}{6} + 4\pi = \frac{25\pi}{6} $$
   $$ \theta = \frac{5\pi}{6} + 4\pi = \frac{29\pi}{6} $$

### Summary

- **General Formulas:**
  $$
  \theta = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2k\pi \quad \text{where } k \text{ is an integer}
  $$
  
- **Six Specific Solutions:**
  $$
  \theta = \frac{\pi}{6}, \quad \frac{5\pi}{6}, \quad \frac{13\pi}{6}, \quad \frac{17\pi}{6}, \quad \frac{25\pi}{6}, \quad \frac{29\pi}{6}
  $$
The general solutions for $$ \sin \theta = \frac{1}{2} $$ are: $$ \theta = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2k\pi $$ where $$ k $$ is any integer. Six specific solutions are: $$ \theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{13\pi}{6}, \frac{17\pi}{6}, \frac{25\pi}{6}, \frac{29\pi}{6} $$

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