Consider the curve C parametrized by

(x, y) = (cos (5t), sin(−4t)), for −π ≤ t ≤ 2pi

How would I go about finding how many times the curve is traversed? Also, how would I find out the radius and center of the circle is? I don't need answers I just need explanations on how to go about doing this.

Thanks for your help!

x has a period of 2pi/5

y has a period of pi/2
The LCM is 2pi, during which time x has gone around 5 times and y has gone around 4 times. The entire Lissajou figure will have been completed once.

so, adding another pi means y goes around twice more, but x only 2.5 times more. You can get a feel for this by looking at the plot at

https://www.wolframalpha.com/input/?i=plot+x%3Dcos%285t%29%2C+y%3Dsin%28-4t%29+for+-pi+%3C%3D+t+%3C%3D+9pi%2F10

You can see that increasing the interval to [-pi,pi] will complete the curve one time

To determine how many times the curve is traversed, you need to find the number of complete cycles made by both the x and y components of the parametric equation.

1. Start by looking at the x-component of the parametric equation, which is cos(5t). The cosine function completes one full cycle (goes from its maximum value to its minimum value and back to the maximum value) in 2π radians. Therefore, the x-component completes a full cycle every 2π/5 radians.

2. Next, analyze the y-component of the parametric equation, which is sin(-4t). The sine function also completes one full cycle in 2π radians. However, note that there is a negative sign before the angle, which means the graph is inverted along the x-axis. This does not affect the number of cycles but changes the orientation of the curve.

3. To determine how many times the curve is traversed, you need to find the least common multiple (LCM) of the cycles of the x and y components. The LCM represents the point where both components complete an integer number of cycles simultaneously. In this case, you would find the LCM of 2π/5 and 2π.

For finding the radius and center of the circle described by the parametric equation:

1. The parametric equation (x, y) = (cos(5t), sin(-4t)) represents a parametric equation of a curve known as a Lissajous curve. It is not necessarily a circle.

2. However, if the frequencies of the cosine and sine functions are proportional (meaning the ratio of their frequencies is a rational number), the resulting parametric equation will trace out a closed loop that could resemble a circle. In this case, the frequency ratio is 5/4, which is a rational number.

3. To find the radius and center of the circle, you need to determine the amplitudes or maximum values of the trigonometric functions.

- For the x-component, the maximum value of cosine is 1, so the amplitude is 1.
- For the y-component, the maximum value of sine is also 1, so the amplitude is 1.

Therefore, the radius of the circle (if it is circular) would be 1, and the center would be at the origin (0, 0).

Keep in mind that if the frequency ratio is not a rational number, the resulting curve will not be a circle, and determining the radius and center may not be applicable.

To determine how many times the curve is traversed, you need to analyze the behavior of both x and y as t ranges from -π to 2π. Start by examining the values of x(t) and y(t) at t = -π and t = 2π and observe their behavior as t changes.

To find out the radius and center of the circle, recall that x = cos(5t) and y = sin(-4t). The general equation of a circle in Cartesian coordinates is given by (x-a)^2 + (y-b)^2 = r^2, where (a, b) represents the center of the circle and r represents the radius.

To determine the radius, you can start by finding the minimum and maximum values of x and y on the interval -π ≤ t ≤ 2π. By analyzing the behavior of cos(5t) and sin(-4t), you can identify the maximum and minimum values of x and y, which will correspond to the radius of the circle.

To find the center of the circle, calculate the average of the maximum and minimum values of x and y. The resulting values will represent the coordinates of the center (a, b) of the circle.

By following these steps, you can determine how many times the curve is traversed and find the radius and center of the circle formed by the parametrized curve.