To draw the graph of the rose curve defined by the equation r = 4 cos(2θ), we need to understand the parameters involved in this polar equation.
The rose curve has a symmetric pattern and its number of petals is determined by the coefficient of θ. In this case, the coefficient is 2. Since this is even, the number of petals will be 2n, which implies we will have 4 petals for this particular rose curve.
Here’s how you can plot the curve:
- Set up the polar coordinates: You will need to calculate the values of r for different angles θ, typically from 0 to 2π.
- Calculate points: For various angles—0, π/6, π/4, π/3, π/2, etc.—substitute into the equation to find the corresponding r values.
- For example:
– When θ = 0, r = 4 cos(0) = 4
– When θ = π/4, r = 4 cos(π/2) = 0
– When θ = π/2, r = 4 cos(π) = -4 (which reflects to 4 in the opposite direction)
Continue calculating until you complete a full revolution. After determining about 8-12 point pairs (r, θ), you can begin plotting these points on polar graph paper or using a graphing utility.
Finally, connect the points smoothly to reveal the beautiful rose curve with four symmetrical petals. Ensure you note that the shape repeats and that the petals are evenly spaced.