Given that θ\thetaθ is a small angle measured in radians, approximate the function sin⁡3(θ)⋅sin⁡(2⋅θ)−3sin⁡(2⋅θ)⋅sin⁡2(4⋅θ)+sin⁡(12⋅θ)+θ⋅sin⁡(θ)sin⁡(2⋅θ),\frac{\sin^3(\theta)\cdot\sin(2\cdot\theta)-3\sin(2\cdot\theta)\cdot\sin^2(4\cdot\theta)+\sin\left(\frac{1}{2}\cdot\theta\right)+\theta\cdot\sin(\theta)}{\sin(2\cdot\theta)},sin(2⋅θ)sin3(θ)⋅sin(2⋅θ)−3sin(2⋅θ)⋅sin2(4⋅θ)+sin(21​⋅θ)+θ⋅sin(θ)​, expressing your answer in terms of θ\thetaθ in a simplified form.

We are asked to approximate and simplify the following expression for small \( \theta \), which is measured in radians:

\[
\frac{\sin^3(\theta)\cdot\sin(2\cdot\theta)-3\sin(2\cdot\theta)\cdot\sin^2(4\cdot\theta)+\sin\left(\frac{1}{2}\cdot\theta\right)+\theta\cdot\sin(\theta)}{\sin(2\cdot\theta)}
\]

### Step 1: Small Angle Approximation

For small \( \theta \), the following approximations hold:

- \( \sin(\theta) \approx \theta \)
- \( \sin(2\theta) \approx 2\theta \)
- \( \sin(4\theta) \approx 4\theta \)
- \( \sin\left(\frac{1}{2}\theta\right) \approx \frac{1}{2}\theta \)

### Step 2: Approximate Each Term

#### 1. \( \sin^3(\theta) \cdot \sin(2\theta) \)

Using \( \sin(\theta) \approx \theta \) and \( \sin(2\theta) \approx 2\theta \):
\[
\sin^3(\theta) \cdot \sin(2\theta) \approx \theta^3 \cdot 2\theta = 2\theta^4
\]

#### 2. \( 3\sin(2\theta) \cdot \sin^2(4\theta) \)

Using \( \sin(2\theta) \approx 2\theta \) and \( \sin(4\theta) \approx 4\theta \):
\[
\sin^2(4\theta) \approx (4\theta)^2 = 16\theta^2
\]
So:
\[
3\sin(2\theta) \cdot \sin^2(4\theta) \approx 3 \cdot 2\theta \cdot 16\theta^2 = 96\theta^3
\]

#### 3. \( \sin\left(\frac{1}{2}\theta\right) \)

Using \( \sin\left(\frac{1}{2}\theta\right) \approx \frac{1}{2}\theta \):
\[
\sin\left(\frac{1}{2}\theta\right) \approx \frac{1}{2}\theta
\]

#### 4. \( \theta \cdot \sin(\theta) \)

Using \( \sin(\theta) \approx \theta \):
\[
\theta \cdot \sin(\theta) \approx \theta \cdot \theta = \theta^2
\]

### Step 3: Plug into the Expression

Substituting all the approximations into the original expression:

\[
\frac{2\theta^4 - 96\theta^3 + \frac{1}{2}\theta + \theta^2}{2\theta}
\]

### Step 4: Simplify the Expression

First, factor out \( \theta \) from the numerator:

\[
\frac{\theta(2\theta^3 - 96\theta^2 + \frac{1}{2} + \theta)}{2\theta}
\]

Cancel \( \theta \) from both the numerator and denominator (assuming \( \theta \neq 0 \)):

\[
\frac{2\theta^3 - 96\theta^2 + \frac{1}{2} + \theta}{2}
\]

This simplifies to:

\[
\frac{1}{2}(2\theta^3 - 96\theta^2 + \theta + \frac{1}{2})
\]

### Step 5: Final Expression

The simplified expression is:

\[
\theta^3 - 48\theta^2 + \frac{1}{2}\theta + \frac{1}{4}
\]

Thus, for small \( \theta \), the expression approximates to:

\[
\boxed{\theta^3 - 48\theta^2 + \frac{1}{2}\theta + \frac{1}{4}}
\]

### Do you want details or have any questions?

### 5 Related Questions:
1. How can small-angle approximations be applied to more complex trigonometric functions?
2. What other common approximations are useful for simplifying trigonometric functions?
3. How do higher-order terms affect the accuracy of small-angle approximations?
4. Can this method be applied for larger angles, and what changes if it does?
5. How does this approximation compare to numerical solutions for small \( \theta \)?

### Tip:
When using small-angle approximations, always ensure the angle is sufficiently small (typically less than 0.1 radians) to maintain accuracy.

Given that θ is a small angle measured in radians, approximate the function sin^3(θ)·sin(2θ) - 3sin(2θ)·sin^2(4θ) + sin(12θ) + θ·sin(θ)sin(2θ), expressing your answer in terms of θ in a simplified form.

This problem involves simplifying a complex trigonometric expression using small angle approximations. Explore how trigonometric functions are approximated for small angles, leveraging algebraic manipulation and fundamental trigonometric principles. Suitable for university-level calculus students.

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