Math Problem Statement
Solution
To determine the equation of the tangent line to the graph of at the point , we need to follow these steps:
Step 1: Find the derivative
Since we need the slope of the tangent line, we calculate by differentiating .
Using the identity , we can rewrite as:
Now, differentiating with respect to :
Step 2: Evaluate at
To find the slope of the tangent line at , substitute into :
So, the slope of the tangent line at is .
Step 3: Write the equation of the tangent line
The point-slope form of a line is given by: where is the slope and is the point on the line.
Here, and the point is . Substituting these values:
Simplifying:
Final Answer
The equation of the tangent line is:
Would you like further explanation on any steps?
Here are five related questions that might deepen your understanding:
- How does using trigonometric identities help simplify derivative calculations?
- Why do we use the point-slope form to find the equation of a tangent line?
- What is the significance of the derivative in finding tangent lines to a curve?
- How would the equation of the tangent line change if evaluated at a different point?
- 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
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