Which is the polar form of the parametric equations x = 5 cos θ and y = 5 sin θ?

Question

(1 point)
Responses

r = 5θ
r = 5 θ

r = 5
r = 5

r = 25 cos θ sin θ
r = 25 cos θ sin θ

r equals 25 cosine squared theta plus 25 sine squared theta

Answer 1

The given parametric equations are \( x = 5 \cos \theta \) and \( y = 5 \sin \theta \).

To convert these equations into polar form, we can use the relationships between Cartesian coordinates (x, y) and polar coordinates (r, θ):

Here, \( x = r \cos \theta \) and \( y = r \sin \theta \).

From the equations given:

1. \( x = 5 \cos \theta \) implies \( r \cos \theta = 5 \),
2. \( y = 5 \sin \theta \) implies \( r \sin \theta = 5 \).

Now, we can express \( r \):

To eliminate θ, we can find r using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} = \sqrt{(5 \cos \theta)^2 + (5 \sin \theta)^2} \] \[ = \sqrt{25 \cos^2 \theta + 25 \sin^2 \theta} = \sqrt{25(\cos^2 \theta + \sin^2 \theta)} = \sqrt{25} = 5 \]

Thus, the polar form of the given parametric equations is \( r = 5 \).

So, the correct answer is:

r = 5