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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 \( \left( \frac{\pi}{2}, 0 \right) \), follow these steps:

### Step 1: Differentiate \( F(x) \)
We need to find the derivative of \( F(x) = 38 \sin(x) \cos(x) \) to get the slope of the tangent line at \( x = \frac{\pi}{2} \).

Using the double-angle identity for trigonometric functions, \( \sin(2x) = 2 \sin(x) \cos(x) \), we can rewrite \( F(x) \) as:
\[
F(x) = 38 \sin(x) \cos(x) = 19 \sin(2x)
\]

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

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

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

### Step 3: Use the Point-Slope Form of the Line
The tangent line passes through \( \left( \frac{\pi}{2}, 0 \right) \) with a slope of \( -38 \). Using the point-slope form \( y - y_1 = m(x - x_1) \), where \( m = -38 \), \( x_1 = \frac{\pi}{2} \), and \( y_1 = 0 \), we get:
\[
y - 0 = -38 \left( x - \frac{\pi}{2} \right)
\]

Simplify this to get the equation of the tangent line:
\[
y = -38x + 19\pi
\]

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

Would you like more details or have any questions about this solution?

Here are some related questions:
1. How would we find the tangent line if \( F(x) \) was a different trigonometric function?
2. Can the point-slope formula be used for any type of function?
3. What if the point given was not on the curve? How would we determine this?
4. How does the double-angle identity simplify differentiating trigonometric functions?
5. Why is it important to rewrite functions using identities when differentiating?

**Tip:** Always look for trigonometric identities that can simplify expressions before differentiating; it often makes the calculation easier.

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 determine the equation of the tangent line to F(x) = 38 sin(x) cos(x) at the point (π/2, 0). This problem covers differentiation techniques, using trigonometric identities, and applying the point-slope form of a line. Suitable for high school students in Grades 11-12.

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