Sketch a well labeled polar curve for r=6theta

To sketch the polar curve for \( r = 6\theta \), follow these steps:

1. **Set up the polar equation**: 
   The equation is \( r = 6\theta \), where \( r \) represents the radius from the origin, and \( \theta \) is the angle (in radians).

2. **Understand the behavior**:
   - As \( \theta \) increases, \( r \) increases proportionally, forming a spiral. This is a form of an **Archimedean spiral**, where each complete rotation adds to the distance from the origin.

3. **Calculate specific points**:
   Choose several values of \( \theta \) to see how \( r \) changes:
   - \( \theta = 0 \): \( r = 6 \times 0 = 0 \)
   - \( \theta = \frac{\pi}{6} \): \( r = 6 \times \frac{\pi}{6} = \pi \)
   - \( \theta = \frac{\pi}{4} \): \( r = 6 \times \frac{\pi}{4} = \frac{3\pi}{2} \)
   - \( \theta = \frac{\pi}{2} \): \( r = 6 \times \frac{\pi}{2} = 3\pi \)
   - \( \theta = \pi \): \( r = 6\pi \)
   - Continue similarly for higher values of \( \theta \).

4. **Plot the points and sketch the spiral**:
   - Plot the points found in step 3 on a polar graph.
   - Connect these points smoothly to form the spiral shape that starts at the origin and moves outward as \( \theta \) increases.

5. **Labeling**:
   - Mark the origin point.
   - Label key angles (\( \theta = 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{2}, \pi \)) and the corresponding \( r \)-values.
   - Show the curve as a continuous spiral expanding outward.

This curve is often called an Archimedean spiral because \( r \) increases linearly with \( \theta \), rather than exponentially.

Would you like further details, or do you have any questions?

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### Related Questions:
1. What is an Archimedean spiral, and how does it differ from a logarithmic spiral?
2. How do polar coordinates differ from Cartesian coordinates?
3. What are some applications of polar graphs in real-world situations?
4. How would the graph of \( r = 3\theta \) compare to \( r = 6\theta \)?
5. What would be the effect if we replaced \( \theta \) with \( \theta + \pi \) in \( r = 6\theta \)?

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**Tip**: When plotting polar curves, always start by evaluating \( r \) at several values of \( \theta \) to understand the shape of the curve better.

Learn to sketch the polar curve r = 6θ, a type of Archimedean spiral, with step-by-step instructions on plotting key points and understanding the relationship between radius and angle in polar coordinates. This guide is ideal for high school students in Grades 11-12.

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