Drawing the Rose Curve: r = 4 cos(2θ)
The rose curve defined by the equation r = 4 cos(2θ) is a fascinating polar graph that showcases a petal structure. Here’s how to draw this curve step-by-step:
Step 1: Understanding the Parameters
- r = the distance from the origin to the curve.
- θ = the angle in radians.
- The coefficient of cos determines the number of petals. In this case, with 2 as the multiplier of θ, we expect 4 petals.
Step 2: Setup the Polar Coordinate System
On a piece of graph paper or using a graphing tool, set up polar coordinates:
- Draw a central point, which will be the origin (0, 0).
- Mark out angles at regular intervals (for example, every 30° or 45°).
Step 3: Calculate Values
For several values of θ (from 0 to 2π), calculate r using the equation:
- For θ = 0: r = 4 cos(0) = 4
- For θ = π/4: r = 4 cos(π/2) = 0
- For θ = π/2: r = 4 cos(π) = -4 (this means moving in the opposite direction)
- Continue calculating for θ = 3π/4, π, 5π/4, 3π/2, and so on.
Step 4: Plot the Points
After calculating r for a range of θ values:
- Plot each (r, θ) point on your polar grid.
- Remember to account for negative r values by plotting the point in the opposite direction.
Step 5: Connect the Dots
Once the points are plotted:
- Connect the dots smoothly to outline the petals.
- You should see a beautiful rose curve with 4 symmetrical petals.
Conclusion
You’ve successfully drawn the rose curve r = 4 cos(2θ)! This curve is not only mathematically intriguing but also visually appealing, making it a great subject for exploration in polar coordinates.