Which of the following is the polar equation for the Cartesian equation y = 1? (15 points)
A) r csc θ = 1
B) r sec θ = 1
C) r sin θ = 1
D) r cos θ = 1
well, y = r sinθ, so ...
To find the polar equation for the Cartesian equation y = 1, we can use the conversion formulas between polar and Cartesian coordinates.
The Cartesian equation y = 1 represents a horizontal line passing through the point (0, 1) on the Cartesian plane.
In polar coordinates, the point (0, 1) can be represented as a point on the positive y-axis.
Since the point lies on the positive y-axis, we can see that the distance r from the origin to the point is 1 unit, and the angle θ between the positive x-axis and the line connecting the origin and the point is 90 degrees or π/2 radians.
Therefore, the polar equation for y = 1 is: r sin θ = 1.
So, the correct answer is C) r sin θ = 1.
To determine the polar equation for the given Cartesian equation y = 1, we need to express y in terms of the polar coordinate system.
In the Cartesian coordinate system, the equation y = 1 represents a horizontal line passing through all points on the y-axis with a constant y-coordinate of 1.
In polar coordinates, the conversion from Cartesian coordinates to polar coordinates can be made using the following relationships:
x = r * cos(θ)
y = r * sin(θ)
Since y = 1, we can substitute this value into the polar representation, which gives us:
r * sin(θ) = 1
Therefore, the polar equation for the Cartesian equation y = 1 is option C) r sin θ = 1.