Graph the parametric equations:
x=cos(t) y=sin(t) z=cos(2t)
How would I go about solving this problem? thanks!
well, you know that the x-y part is just a circle. So, as x-y moves around the circle, z bobs up and down twice. It's saddle-shaped.
Visit wolframalpha.com and type
plot x=cos(t), y=sin(t), z=cos(t)
To graph the parametric equations x = cos(t), y = sin(t), and z = cos(2t), follow these steps:
1. Determine the range of values for the parameter t that you want to use. For example, you could choose a range such as t ∈ [0, 2π] if you want to graph one full period.
2. Choose several values of t within your chosen range. For example, you could choose t = 0, π/4, π/2, π, 3π/4, and 2π.
3. Substitute each chosen value of t into the parametric equations to find the corresponding (x, y, z) coordinates. For example, when t = 0, x = cos(0) = 1, y = sin(0) = 0, and z = cos(2*0) = 1.
4. Repeat the substitution process for each chosen value of t to get a list of (x, y, z) coordinates.
5. Plot the obtained coordinates on a three-dimensional coordinate system. Connect the plotted points smoothly to visualize the graph of the parametric equations.
6. Optional: If you want to enhance your graph, you can increase the density of points by choosing more values of t within your specified range. This will make the curve smoother.
Remember that the choice of the range for t and the density of sampled points will affect the appearance of the graph. Adjust these parameters as needed to accurately represent the desired curve.
To graph the parametric equations x = cos(t), y = sin(t), and z = cos(2t), you need to plot the corresponding points in three-dimensional space. Here's how you can solve this problem:
1. Choose a range for the parameter t. Let's say you want to graph the equations for t values ranging from 0 to 2π.
2. Calculate the values of x, y, and z for different values of t within the chosen range. For example, you can calculate x, y, and z for t = 0, t = π/4, t = π/2, t = 3π/4, and so on.
3. Once you have the values of x, y, and z, plot these points in a three-dimensional coordinate system.
4. Connect the plotted points to get a smooth curve, which represents the graph of the parametric equations.
To make it easier to visualize, you can also use a graphing software or an online graphing tool that supports three-dimensional graphs. Simply input the equations x = cos(t), y = sin(t), and z = cos(2t) and specify the range for t. The software will generate the graph for you.
Remember that the parameter t determines the position on the curve, so different values of t will give you different points on the graph. By varying t within the chosen range, you will be able to see the full representation of the curve in three-dimensional space.