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We are tasked with finding the equation of the tangent to the graph of the function \( y = \arcsin\left(\frac{x}{2}\right) \) at the origin, i.e., the point \( (0, 0) \).

### Steps to find the tangent equation:

1. **Differentiate the function:**
   The equation of the function is \( y = \arcsin\left(\frac{x}{2}\right) \). We need to differentiate it with respect to \( x \) to find the slope of the tangent line.

   Using the chain rule:
   \[
   \frac{dy}{dx} = \frac{1}{\sqrt{1 - \left(\frac{x}{2}\right)^2}} \cdot \frac{1}{2}
   \]
   Simplifying:
   \[
   \frac{dy}{dx} = \frac{1}{2\sqrt{1 - \frac{x^2}{4}}}
   \]

2. **Evaluate the derivative at the origin:**
   To find the slope of the tangent line at the origin \( x = 0 \), substitute \( x = 0 \) into the derivative:
   \[
   \frac{dy}{dx} \bigg|_{x = 0} = \frac{1}{2\sqrt{1 - \frac{0^2}{4}}} = \frac{1}{2}
   \]
   So, the slope of the tangent line at the origin is \( \frac{1}{2} \).

3. **Equation of the tangent line:**
   The general equation of a tangent line at a point \( (x_0, y_0) \) is given by:
   \[
   y - y_0 = m(x - x_0)
   \]
   Here, \( m = \frac{1}{2} \), \( x_0 = 0 \), and \( y_0 = 0 \). Therefore, the equation of the tangent line at the origin is:
   \[
   y = \frac{1}{2}x
   \]

### Final Answer:
The equation of the tangent line to the graph of \( y = \arcsin\left(\frac{x}{2}\right) \) at the origin is:
\[
y = \frac{1}{2}x
\]

Would you like further clarification or details on any part of the solution?

Here are five related questions:
1. How is the derivative of the inverse sine function derived?
2. What is the significance of the chain rule in this problem?
3. How can you determine if a tangent line is horizontal at a point?
4. Can we find the tangent at any other point on this function?
5. How does the arc sine function behave for values of \( x \) close to \( 2 \)?

**Tip:** Always verify the domain of inverse trigonometric functions to avoid evaluating them at points where they are undefined.

An equation for a tangent to the graph of y = arcsin(x/2) at the origin is

Find the equation of the tangent line to the graph of y = arcsin(x/2) at the origin. This solution involves differentiating the inverse trigonometric function using the chain rule, evaluating the slope at the origin, and determining the equation of the tangent line. Suitable for students in Grades 11-12.

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