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.