To find the third-order Maclaurin polynomial $T_3(x)$ for $f(x) = \sin(3x)\cos(x)$, we can proceed as follows:

Step 1: Use Trigonometric Identity

First, use the trigonometric identity: $\sin(A)\cos(B) = \frac{1}{2} \left( \sin(A+B) + \sin(A-B) \right).$ Substitute $A = 3x$ and $B = x$: $\sin(3x)\cos(x) = \frac{1}{2} \left( \sin(4x) + \sin(2x) \right).$

Thus, $f(x) = \frac{1}{2} \left( \sin(4x) + \sin(2x) \right).$

Step 2: Expand $\sin(4x)$ and $\sin(2x)$ as Taylor Series

The Maclaurin series for $\sin(x)$ is: $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots.$

For $\sin(4x)$: $\sin(4x) = 4x - \frac{(4x)^3}{3!} + \cdots = 4x - \frac{64x^3}{6} + \cdots = 4x - \frac{32x^3}{3} + \cdots.$

For $\sin(2x)$: $\sin(2x) = 2x - \frac{(2x)^3}{3!} + \cdots = 2x - \frac{8x^3}{6} + \cdots = 2x - \frac{4x^3}{3} + \cdots.$

Step 3: Combine and Simplify

Now, substitute these into $f(x)$: $f(x) = \frac{1}{2} \left( \sin(4x) + \sin(2x) \right),$ $f(x) = \frac{1}{2} \left[ \left( 4x - \frac{32x^3}{3} + \cdots \right) + \left( 2x - \frac{4x^3}{3} + \cdots \right) \right].$

Simplify the terms:

• Combine the linear terms: $4x + 2x = 6x$,
• Combine the cubic terms: $-\frac{32x^3}{3} - \frac{4x^3}{3} = -\frac{36x^3}{3} = -12x^3$.

Thus, $f(x) = \frac{1}{2} \left( 6x - 12x^3 + \cdots \right),$ $f(x) = 3x - 6x^3 + \cdots.$

Step 4: Third-Order Maclaurin Polynomial

The third-order Maclaurin polynomial is: $T_3(x) = 3x - 6x^3.$

Would you like more details or a verification? Here are five follow-up questions for deeper exploration:

1. How would the result change if higher-order terms were included in the expansion?
2. Could this solution be verified using direct differentiation?
3. How does the Maclaurin series for cosine influence similar expansions?
4. What happens if we use another trigonometric identity for simplification?
5. Can this method be generalized for other combinations of trigonometric functions?

Tip: Always check if identities can simplify your initial function before expanding—this can save significant time in calculations!