Math Problem Statement

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).

Solution

To determine the equation of the tangent line to the graph of F(x)=38sin(x)cos(x)F(x) = 38 \sin(x) \cos(x) at the point (π2,0)\left(\frac{\pi}{2}, 0\right), we need to follow these steps:

Step 1: Find the derivative F(x)F'(x)

Since we need the slope of the tangent line, we calculate F(x)F'(x) by differentiating F(x)=38sin(x)cos(x)F(x) = 38 \sin(x) \cos(x).

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

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

Step 2: Evaluate F(x)F'(x) at x=π2x = \frac{\pi}{2}

To find the slope of the tangent line at x=π2x = \frac{\pi}{2}, substitute x=π2x = \frac{\pi}{2} into F(x)F'(x): F(π2)=38cos(2π2)=38cos(π)=38(1)=38F'\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=π2x = \frac{\pi}{2} is 38-38.

Step 3: Write the equation of the tangent line

The point-slope form of a line is given by: yy1=m(xx1)y - y_1 = m(x - x_1) where mm is the slope and (x1,y1)(x_1, y_1) is the point on the line.

Here, m=38m = -38 and the point is (π2,0)\left(\frac{\pi}{2}, 0\right). Substituting these values: y0=38(xπ2)y - 0 = -38 \left(x - \frac{\pi}{2}\right)

Simplifying: y=38x+19πy = -38x + 19\pi

Final Answer

The equation of the tangent line is: y=38x+19πy = -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.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Differentiation
Tangent Line
Trigonometric Identities

Formulas

Derivative of sin(x)cos(x): F'(x) = 38 cos(2x)
Point-slope form of a line: y - y1 = m(x - x1)
Trigonometric identity: sin(2x) = 2 sin(x) cos(x)

Theorems

Derivative Rule
Trigonometric Identity
Point-Slope Form of a Line

Suitable Grade Level

Grade 12 or First-year Calculus