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Sketch the graph of r equals 6 cos left parenthesis 5 theta right parenthesis.

The polar equation \( r = 6 \cos(5\theta) \) describes a rose curve with petals, specifically due to the cosine function and the coefficient of \( \theta \) being an odd integer (5). Here's the breakdown of the graph characteristics:

1. **Petal Count**: Since the coefficient of \( \theta \) is 5, this rose curve will have exactly 5 petals.
2. **Petal Length**: The maximum value of \( r \) (which is the amplitude, here 6) determines the length of each petal. Thus, each petal will reach out to a radius of 6 units.
3. **Symmetry and Orientation**: Because it's a cosine function, the petals are oriented symmetrically along the polar axis.

In terms of the images you provided:
- **Top-left graph**: This image has 10 petals, which would correspond to an equation with a coefficient of 10 in \( \cos(k\theta) \) or \( \sin(k\theta) \), where \( k \) is even.
- **Top-right graph**: This has 8 petals, suggesting a similar form, likely \( r = a \cos(4\theta) \) or \( r = a \sin(4\theta) \).
- **Bottom-left and bottom-right graphs**: These images display 5 petals each, matching the characteristics of \( r = 6 \cos(5\theta) \).

Thus, the **bottom two graphs** are likely the correct sketches for \( r = 6 \cos(5\theta) \). The orientation of the petals aligns with the cosine function, with each petal extending outward in a symmetric pattern. 

Would you like further details on rose curve properties or the step-by-step plotting process?

Here are some related questions to consider:
1. How would the graph change if \( \cos \) was replaced by \( \sin \)?
2. What happens if the amplitude changes, such as \( r = 8 \cos(5\theta) \)?
3. How do even values for the coefficient of \( \theta \) affect the petal count?
4. Can the rose curve formula have negative values for \( r \), and how does that affect the graph?
5. How would a rose curve look if plotted in Cartesian coordinates?

**Tip**: For polar equations with rose curves, the petal count doubles if \( k \) is even and stays the same if \( k \) is odd.

Learn how to sketch the rose curve defined by the polar equation r = 6 cos(5θ). This problem explores the properties of rose curves, with a focus on petal count and length for odd values of k, symmetry, and the role of cosine in polar equations. Suitable for students in Grades 10-12.

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