Which is the polar form of the parametric equations x=4 cos theta and y=4 sun theta?
AAAaannndd the bot gets it wrong yet again!
The polar form is just r=4
the rectangular form is x^2 + y^2 = 16
No exponentials and complex numbers are involved.
You are correct. Apologies for the mistake in my previous response.
The polar form of the parametric equations x = 4 cos(theta) and y = 4 sin(theta) is just r = 4, which represents a circle centered at the origin with radius 4.
The rectangular form can be found by using the Pythagorean identity as:
x^2 + y^2 = (4 cos(theta))^2 + (4 sin(theta))^2
x^2 + y^2 = 16(cos^2(theta) + sin^2(theta))
x^2 + y^2 = 16
This represents a circle centered at the origin with radius 4, as expected. Once again, apologies for any confusion caused by my earlier response.
Thanks oobleck !
You're welcome! If you have any other questions, feel free to ask.
To find the polar form of the given parametric equations x = 4cos(theta) and y = 4sin(theta), we can make use of the relationships between Cartesian (x, y) and polar (r, theta) coordinates.
In the case of the parametric equations, we have x = 4cos(theta) and y = 4sin(theta). We need to convert these equations into polar form, which means expressing them in terms of r and theta.
First, let's recall the relationship between Cartesian and polar coordinates:
x = r * cos(theta)
y = r * sin(theta)
Now, we can compare these equations to the given parametric equations:
x = 4cos(theta) --> x = r * cos(theta)
y = 4sin(theta) --> y = r * sin(theta)
Comparing the equations, we can see that r = 4 in both cases.
Therefore, the polar form of the given parametric equations is r = 4.
Simply put, the polar form of the equations x = 4cos(theta) and y = 4sin(theta) is r = 4.
In polar form, x = r cos(theta) and y = r sin(theta), where r is the distance from the origin and theta is the angle from the positive x-axis to a point (x,y) in standard position.
To find the polar form of x = 4 cos(theta) and y = 4 sin(theta), we can use the Pythagorean identity:
r^2 = x^2 + y^2
Substituting the given equations:
r^2 = (4 cos(theta))^2 + (4 sin(theta))^2
Simplifying:
r^2 = 16(cos^2(theta) + sin^2(theta))
r^2 = 16
Taking the square root:
r = ± 4
Since x and y are both positive, we can take r = 4.
Now, we can write the polar form as:
r cos(theta) + i r sin(theta) = 4 cos(theta) + 4i sin(theta)
Or, using the exponential form of complex numbers:
4e^(i theta)
Therefore, the polar form is:
r = 4
θ = arctan(y/x) = arctan(4 sin(theta)/4 cos(theta)) = arctan(tan(theta)) = theta
So, the polar form of x=4 cos theta and y=4 sin theta is:
r = 4, θ = theta
Or, equivalently:
4e^(i theta)