Identify the parametric equations that represent the same path as the following parametric equations.
x(t)=2cos2t
y(t)=sin3t
a. x(t)=2cos2t
y(t)=sin6t
b. x(t)=4cos4t
y(t)=sin6t
c. x(t)=2cos4t
y(t)=sin6t
d. x(t)=4cos2t
y(t)=2sin3t
Looks like (c) to me. Just set u=2t and you have
x(t)=2cos2u
y(t)=sin3u
The names have been changed to protect the innocent parameters.
Well, to find the equivalent parametric equations, we should match the coefficients of the trigonometric functions.
In the given parametric equations, x(t) = 2cos(2t) and y(t) = sin(3t).
In option a, the coefficient of sin(6t) matches with the original y(t) equation (sin(3t)), but the coefficient of cos(2t) doesn't match with the original x(t) equation (2cos(2t)). So, option a is not the correct choice.
In option b, both the coefficients match with the original x(t) and y(t) equations. So, option b is a possible answer.
In option c, while the coefficient of sin(6t) matches, the coefficient of cos(4t) doesn't match with the original x(t) equation. So, option c is not correct.
In option d, the coefficient of cos(2t) matches but the coefficient of 2sin(3t) doesn't match with the original y(t) equation. So, option d is not the answer.
Therefore, the correct option is b. x(t) = 4cos(4t) and y(t) = sin(6t) represent the same path as the given parametric equations.
To identify the parametric equations that represent the same path as x(t) = 2cos(2t) and y(t) = sin(3t), we need to consider the factors multiplying t within the cosine and sine functions.
In this case, the period of the cosine function is determined by the coefficient of t, while the period of the sine function is determined by the coefficient of t as well. The frequency (number of complete cycles) of the cosine and sine functions can be found by dividing the coefficient of t by 2π.
Given that the coefficient of t in x(t) is 2, it indicates a period of 2π/2 = π. Thus, the frequency of the cosine is 1/π.
For y(t), the coefficient of t is 3, which indicates a period of 2π/3. Therefore, the frequency of the sine function is 1/(2π/3) = 3/2π.
Now, let us evaluate the given options:
a. x(t) = 2cos(2t), y(t) = sin(6t)
In this choice, the cosine has a frequency of 1/2π and the sine has a frequency of 3/2π, which matches the original parametric equations. Hence, option (a) is a correct choice.
b. x(t) = 4cos(4t), y(t) = sin(6t)
Here, the cosine has a frequency of 1/4π, which does not match the original frequency. Thus, option (b) is incorrect.
c. x(t) = 2cos(4t), y(t) = sin(6t)
In this case, the cosine has a frequency of 1/4π, which is different from the original frequency. Therefore, option (c) is incorrect.
d. x(t) = 4cos(2t), y(t) = 2sin(3t)
Here, the cosine has a frequency of 1/2π, which matches the original frequency. However, the sine has a frequency of 3/2π, which is different. Therefore, option (d) is incorrect.
In conclusion, the correct choice is option (a):
x(t) = 2cos(2t)
y(t) = sin(6t)
To identify the parametric equations that represent the same path as the given parametric equations, we need to compare the functions for x(t) and y(t) individually.
Let's break it down:
Given parametric equations:
x(t) = 2cos(2t)
y(t) = sin(3t)
1. Option a:
x(t) = 2cos(2t)
y(t) = sin(6t)
To check if it represents the same path, let's compare the coefficients of t in both cases:
For x(t), the coefficient of t is 2 in both cases, so it is the same.
For y(t), the coefficient of t is 3 in the given equations but 6 in option a. They are different.
Since the coefficients of t are not the same for y(t), option a does not represent the same path.
2. Option b:
x(t) = 4cos(4t)
y(t) = sin(6t)
Comparing the coefficients of t:
For x(t), the coefficient of t is 4 in option b, while it is 2 in the given equations. They are different.
For y(t), the coefficient of t is 3 in the given equations, but there is no t term in option b.
Since the coefficients of t are not the same for x(t) and y(t), option b does not represent the same path.
3. Option c:
x(t) = 2cos(4t)
y(t) = sin(6t)
Comparing the coefficients of t:
For x(t), the coefficient of t is 4 in both cases, so it is the same.
For y(t), the coefficient of t is 3 in the given equations, while there is no t term in option c.
Since the coefficients of t are not the same for y(t), option c does not represent the same path.
4. Option d:
x(t) = 4cos(2t)
y(t) = 2sin(3t)
Comparing the coefficients of t:
For x(t), the coefficient of t is 2 in both cases, so it is the same.
For y(t), the coefficient of t is 3 in the given equations, while there is no t term in option d.
Since the coefficients of t are not the same for y(t), option d does not represent the same path.
Therefore, none of the given options represent the same path as the given parametric equations x(t)=2cos(2t) and y(t)=sin(3t).