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To determine the equation of the tangent line to the graph of \( F(x) = 38 \sin(x) \cos(x) \) at the point \( \left(\frac{\pi}{2}, 0\right) \), we need to follow these steps:

### Step 1: Find the derivative \( F'(x) \)
Since we need the slope of the tangent line, we calculate \( F'(x) \) by differentiating \( F(x) = 38 \sin(x) \cos(x) \).

Using the identity \( \sin(2x) = 2 \sin(x) \cos(x) \), we can rewrite \( F(x) \) as:
\[
F(x) = 38 \sin(x) \cos(x) = 19 \sin(2x)
\]

Now, differentiating \( F(x) = 19 \sin(2x) \) with respect to \( x \):
\[
F'(x) = 19 \cdot \cos(2x) \cdot 2 = 38 \cos(2x)
\]

### Step 2: Evaluate \( F'(x) \) at \( x = \frac{\pi}{2} \)
To find the slope of the tangent line at \( x = \frac{\pi}{2} \), substitute \( x = \frac{\pi}{2} \) into \( F'(x) \):
\[
F'\left(\frac{\pi}{2}\right) = 38 \cos\left(2 \cdot \frac{\pi}{2}\right) = 38 \cos(\pi) = 38 \cdot (-1) = -38
\]

So, the slope of the tangent line at \( x = \frac{\pi}{2} \) is \( -38 \).

### Step 3: Write the equation of the tangent line
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is the point on the line.

Here, \( m = -38 \) and the point is \( \left(\frac{\pi}{2}, 0\right) \). Substituting these values:
\[
y - 0 = -38 \left(x - \frac{\pi}{2}\right)
\]

Simplifying:
\[
y = -38x + 19\pi
\]

### Final Answer
The equation of the tangent line is:
\[
y = -38x + 19\pi
\]

Would you like further explanation on any steps?

Here are five related questions that might deepen your understanding:
1. How does using trigonometric identities help simplify derivative calculations?
2. Why do we use the point-slope form to find the equation of a tangent line?
3. What is the significance of the derivative in finding tangent lines to a curve?
4. How would the equation of the tangent line change if evaluated at a different point?
5. What are other methods to find the slope of the tangent if the function was more complex?

**Tip:** Always look for trigonometric identities that can simplify expressions before differentiating, as it can make the calculus easier to handle.

Determine the equation of the line y tangent to the graph of F(x) = 38 sin(x) cos(x) at the point (π/2, 0).

Learn how to find the tangent line to the function F(x) = 38 sin(x) cos(x) at the point (π/2, 0). This problem covers differentiation, trigonometric identities, and applying the point-slope form. Perfect for students in Grade 12 or introductory calculus.

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