To solve the equation \( \sin x = \pm \frac{1}{4} \), we need to find the angles \( x \) such that the sine of \( x \) equals either \( \frac{1}{4} \) or \( -\frac{1}{4} \).

### Step-by-Step Solution:

1. **General Equation**: 
   The general solution for \( \sin x = a \) is:
   \[
   x = \arcsin(a) + 2k\pi \quad \text{or} \quad x = \pi - \arcsin(a) + 2k\pi
   \]
   where \( k \) is any integer, representing the periodicity of the sine function.

2. **Case 1: \( \sin x = \frac{1}{4} \)**
   \[
   x_1 = \arcsin\left(\frac{1}{4}\right) + 2k\pi
   \]
   \[
   x_2 = \pi - \arcsin\left(\frac{1}{4}\right) + 2k\pi
   \]
   where \( \arcsin\left(\frac{1}{4}\right) \) is the angle whose sine is \( \frac{1}{4} \).

3. **Case 2: \( \sin x = -\frac{1}{4} \)**
   \[
   x_3 = \arcsin\left(-\frac{1}{4}\right) + 2k\pi = -\arcsin\left(\frac{1}{4}\right) + 2k\pi
   \]
   \[
   x_4 = \pi - \arcsin\left(-\frac{1}{4}\right) + 2k\pi = \pi + \arcsin\left(\frac{1}{4}\right) + 2k\pi
   \]

### Numerical Approximation:
Using a calculator to find \( \arcsin\left(\frac{1}{4}\right) \), we get approximately:
\[
\arcsin\left(\frac{1}{4}\right) \approx 0.2527 \, \text{radians}
\]

So, the general solutions are:
- For \( \sin x = \frac{1}{4} \):
  \[
  x_1 \approx 0.2527 + 2k\pi
  \]
  \[
  x_2 \approx \pi - 0.2527 + 2k\pi \approx 2.8889 + 2k\pi
  \]

- For \( \sin x = -\frac{1}{4} \):
  \[
  x_3 \approx -0.2527 + 2k\pi
  \]
  \[
  x_4 \approx \pi + 0.2527 + 2k\pi \approx 3.3943 + 2k\pi
  \]

These represent all the solutions for the given equation.

Would you like further details or explanations on any of these steps?

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Here are 5 related questions you could explore:
1. How do you solve \( \cos x = \frac{1}{4} \)?
2. What is the period of the sine function and how does it affect the solution?
3. How would you solve \( \sin(2x) = \pm \frac{1}{4} \)?
4. What are the principal values for inverse sine and cosine functions?
5. How can you express angles in degrees instead of radians?

**Tip**: Always remember that the sine function repeats every \( 2\pi \), which is why the general solution involves adding multiples of \( 2\pi \).

Learn how to solve the trigonometric equation sin(x) = +/- 1/4 with a detailed step-by-step solution. This problem covers key concepts in trigonometry, including the sine function, inverse trigonometric functions, and periodic properties. Suitable for students in Grades 10-12.

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