To determine the equation of the rose curve with 4 petals, we can start by recalling the general form of a rose curve equation in polar coordinates:
- \( r = a \cos(n\theta) \) or \( r = a \sin(n\theta) \)
where:
- \( a \) determines the length of each petal,
- \( n \) determines the number of petals. If \( n \) is even, the number of petals is \( 2n \) (but since you have 4 petals, which is even, this holds true).
In your case, you have 4 petals, so \( n = 2 \). Each petal has a length of 6 units, which means \( a = 6 \).
Because the petals are oriented at an angle of \( \frac{\pi}{4} \) radians from the polar axis, we need to apply a rotation. For a cosine function, this means we write the equation as:
- \( r = 6 \cos(2(\theta - \frac{\pi}{4})) \)
However, from the provided responses, we first want to recognize the forms without transformations. The equations given are structured in terms of sine and cosine directly.
To identify the correct response:
- For \( r = 6 \sin(2\theta) \), the petals are aligned at \( \frac{\pi}{2} \) (90 degrees), but need to be rotated.
- For \( r = 6 \cos(2\theta) \), the petals start aligned to the x-axis (0 degrees) but can rotate to \( \frac{\pi}{4} \).
Since the specific arrangement (angle of \( \frac{\pi}{4} \)) will not affect the selection, examine both:
- If we were to keep \( \cos \), we really only need to modify it to represent the angle correctly. However, of the choices you've presented, the option closest is \( r = 6 \cos(2\theta) \) since it matches 4 petals with a length of 6 units best given the starting criteria without complex transformations.
Thus, the answer is:
\( r = 6 \cos(2\theta) \).