Find the value of theta*θ* in the interval 0, is less than or equal to, theta, is less than or equal to, start fraction, pi, divided by, 2, end fraction0≤*θ*≤2*π*​ that corresponds to the point on the curve r, equals, minus, 4, minus, sine, left parenthesis, 4, theta, right parenthesis, minus, theta, cosine, left parenthesis, theta, right parenthesis*r*=−4−sin(4*θ*)−*θ*cos(*θ*) with the shortest distance from the pole. Round to the nearest thousandth.

To find the value of \(\theta\) in the interval \(0 \leq \theta \leq \frac{\pi}{2}\) that corresponds to the point on the curve \(r = -4 - \sin(4\theta) - \theta \cos(\theta)\) with the shortest distance from the pole (origin), we need to find the minimum value of the function \(r(\theta)\) within the given interval. The distance from the pole is minimized when \(r(\theta)\) is minimized.

### Steps to solve:

1. **Define the function**:
   \[
   r(\theta) = -4 - \sin(4\theta) - \theta \cos(\theta)
   \]

2. **Take the derivative** of \(r(\theta)\) with respect to \(\theta\) to find the critical points:
   \[
   \frac{dr}{d\theta} = -4 \cos(4\theta) - \cos(\theta) + \theta \sin(\theta)
   \]

3. **Set the derivative equal to zero** to find critical points:
   \[
   -4 \cos(4\theta) - \cos(\theta) + \theta \sin(\theta) = 0
   \]
   Solve this equation for \(\theta\).

4. **Evaluate \(r(\theta)\)** at the critical points and the endpoints of the interval \(0 \leq \theta \leq \frac{\pi}{2}\).

5. **Determine the minimum value** of \(r(\theta)\) within the interval by comparing the values obtained in the previous step.

6. **Round \(\theta\)** corresponding to the minimum \(r(\theta)\) to the nearest thousandth.

Let me perform these calculations for you.The value of \(\theta\) in the interval \(0 \leq \theta \leq \frac{\pi}{2}\) that corresponds to the point on the curve with the shortest distance from the pole is approximately \(\theta = 0.438\) radians. The corresponding minimum value of \(r(\theta)\) is approximately \(-5.380\).

Would you like more details or have any questions? Here are five related questions you might consider:

1. How does changing the interval affect the value of \(\theta\) that minimizes \(r(\theta)\)?
2. What is the significance of the function \(r(\theta)\) in polar coordinates?
3. Can you find the second derivative to confirm whether this point is a minimum?
4. How would the solution change if the function \(r(\theta)\) were different?
5. How does the periodic nature of the trigonometric functions affect the results?

**Tip:** When dealing with trigonometric functions, it's useful to visualize the graph to better understand the behavior of the function over the given interval.

Discover how to find the value of theta in the interval 0 ≤ theta ≤ π/2 that corresponds to the point on the curve r = -4 - sin(4θ) - θcos(θ) with the shortest distance from the pole (origin). Learn about polar coordinates, derivatives, and trigonometric functions. Suitable for advanced high school students and undergraduates.

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