convert polar equations to rectangular coordinates.
1.) θ=π
2.) r=6 cos θ
3.) sec θ=2
x=r*cos(è)
y=r*sin(è)
1.)
x=r*cos(è)=r*cos(ð)=r*(-1)= -r
y=r*sin(è)=r*sin(ð)=r*0=0
2.)
x=r*cos(è)=6*cos(è)*cos(è)=6*cos(è)^2
y=r*sin(è)=6*sin(è)*cos(è)=6*(1/2)*sin(2è)=3*sin(2è)
3.)
sec(è)=2
sec(è)=1/cos(è)=2
cos(è)=1/2
sin(è)=sqroot(1-cos(è)^2)
sin(è)=sqroot(1-(1/2)^2)
sin(è)=sqroot(1-1/4)
sin(è)=sqroot(3/4)
sin(è)=sqroot(3)/2
x=r*cos(è)=r/2
y=r*sin(è)=r*sqroot(3)/2
1)y=0, x<0 (the ray)
2)r^2=6r*cos(teta)
x^2+y^2=6x
x^2-6x+9-9+y^2=0
(x-3)^2+y^2=3^2 (the circle)
3)2cos(teta)=1
2r*cos(teta)=r
2x=sqrt(x^2+y^2) (x>0)
4x^2=x^2+y^2
y^2=3x^2
y=(+-)sqrt(3)x (two rays)
To convert polar equations to rectangular coordinates, we can use the following conversions:
1. θ = π:
In polar coordinates, θ represents the angle. Here, θ = π represents a half-circle in the counterclockwise direction. To convert this to rectangular coordinates, we can use the following relations:
x = r * cos(θ)
y = r * sin(θ)
Since θ = π, we can substitute π into the equations:
x = r * cos(π) = r * (-1)
y = r * sin(π) = r * 0
Therefore, the rectangular coordinates for the polar equation θ = π are x = -r and y = 0.
2. r = 6 cos(θ):
In this polar equation, r represents the distance from the origin, and θ represents the angle. To convert this to rectangular coordinates, we can use the following relations:
x = r * cos(θ)
y = r * sin(θ)
For r = 6 cos(θ), we can substitute it into the equations:
x = (6 cos(θ)) * cos(θ) = 6 cos²(θ)
y = (6 cos(θ)) * sin(θ) = 6 cos(θ) * sin(θ)
Therefore, the rectangular coordinates for the polar equation r = 6 cos(θ) are x = 6 cos²(θ) and y = 6 cos(θ) * sin(θ).
3. sec(θ) = 2:
In this equation, sec(θ) represents the reciprocal of cosine, which is equal to 1/cos(θ). To convert this to rectangular coordinates, we can use the following relation:
x = r * cos(θ)
y = r * sin(θ)
Since sec(θ) = 2, we can express it as 1/cos(θ) = 2. By rearranging the equation, we have:
cos(θ) = 1/2
To find the value of θ, we can use the inverse cosine function (also called arccosine or cos^(-1)). So, θ = arccos(1/2).
Substituting this value into the rectangular conversion equations:
x = r * cos(arccos(1/2))
y = r * sin(arccos(1/2))
To simplify further, we need to know the value of r (the distance from the origin), which is not provided in this equation. Without knowing r, we cannot determine the exact rectangular coordinates for the equation sec(θ) = 2.
Therefore, we cannot compute the rectangular coordinates for the polar equation sec(θ) = 2 without additional information.