The letters x and y represent rectangular coordinates. Write the following equation using polar coordinates (r, θ).
Given equation: y=x
I chose: r=cosθ
no, r = cosθ is a circle with radius 1/2 and center at (1/2,0)
y = x
Is just a line with a slope of 1, right?
r sinθ = r cosθ
sinθ = cosθ
tanθ = 1
don't forget your algebra 1.
To rewrite the given equation using polar coordinates, we need to express both x and y in terms of polar coordinates (r, θ).
We know that in rectangular coordinates, x represents the horizontal position and y represents the vertical position. In polar coordinates, r represents the distance from the origin (0, 0) to the point (x, y), and θ represents the angle measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point (x, y).
First, let's find x and y in terms of r and θ. We can use the following relationships:
x = r * cos(θ)
y = r * sin(θ)
Now we can substitute these expressions for x and y in the given equation y = x:
r * sin(θ) = r * cos(θ)
To simplify this equation and express it solely in terms of polar coordinates, we can divide both sides by r:
sin(θ) = cos(θ)
However, this equation does not reduce further and does not match the equation r = cos(θ). Therefore, choosing r = cos(θ) does not accurately represent the equation y = x using polar coordinates.