To graph the parametric equations x = 7 sin(t) sin(7t) and y = 7 cos(t) cos(7t), follow these steps:
1. Understanding the Parametric Equations
The equations define a curve in the XY-plane where the position of each point is determined by the parameter t. The x and y values oscillate between -7 and 7 based on the values of the sine and cosine functions.
2. Choosing a Range for t
First, decide on an appropriate range for t. A complete oscillation for the sine and cosine functions can be observed over the interval [0, 2π] or even longer. Try using t values from 0 to 2π for a basic visualization, or extend to [0, 4π] to see more of the curve.
3. Calculating Points
Next, you can calculate points on the graph. For example:
- For
t = 0:x = 7 sin(0) sin(0) = 0,y = 7 cos(0) cos(0) = 7 - For
t = π/4:x = 7 sin(π/4) sin(7π/4) = -7,y = 7 cos(π/4) cos(7π/4) = 0 - For
t = π/2:x = 7 sin(π/2) sin(7π/2) = 0,y = 7 cos(π/2) cos(7π/2) = -7 - For
t = 3π/4:x = 7 sin(3π/4) sin(7×3π/4) = 7,y = 7 cos(3π/4) cos(7×3π/4) = 0 - For
t = π:x = 7 sin(π) sin(7π) = 0,y = 7 cos(π) cos(7π) = 7
Continue calculating points in this way for the chosen interval.
4. Plotting the Points
Once you have a number of points calculated, you can plot them on graph paper or using graphing software like Desmos or GeoGebra. Connect the points smoothly to visualize the curve.
5. Analyzing the Graph
The resulting graph will have a pattern influenced by both the sin(t) and cos(t) functions, producing an intricate shape as t varies. Expect to see loops and oscillations due to the periodic nature of the sine and cosine functions used in both equations.
Conclusion
Graphing parametric equations can be rewarding, presenting unique shapes and patterns. Experimenting with different intervals for t and observing how the graph changes is an excellent way to develop a deeper understanding of parametric representations!