Which equation is the polar equivalent to the equation y=−(√3/3)x?
θ=π/6
θ=π/3
θ=2π/3
θ=5π/6
To find the polar equivalent of the equation y=−(√3/3)x, we can use the conversion formulas:
r = √(x^2 + y^2)
θ = arctan(y/x)
Substituting the given equation y = - (√3/3)x, we can find the values of r and θ.
First, let's rearrange the equation to solve for x:
y = - (√3/3)x
x = - (√3/3)y
Now, we substitute x and y values into the formulas:
r = √((- (√3/3)y)^2 + y^2)
r = √(3y^2/9 + y^2)
r = √(4y^2/9)
r = (2/3)y
Next, we find the value of θ using the formula:
θ = arctan(y/x)
θ = arctan(y/(- (√3/3)y))
θ = arctan(-√3)
Now we need to compare the obtained value of θ with the given options.
θ=π/6
θ=π/3
θ=2π/3
θ=5π/6
Since none of the given options matches θ = arctan(-√3), there is no direct match for the polar equivalent of the equation y = - (√3/3)x.
To find the polar equivalent of the given equation, we need to convert it from Cartesian coordinates to polar coordinates.
The Cartesian equation y=−(√3/3)x represents a line with a negative slope of -√3/3.
In polar coordinates, the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ) is as follows:
x = r*cos(θ)
y = r*sin(θ)
We can substitute the given equation into the polar coordinate equations:
-(√3/3)x = r*cos(θ)
y = r*sin(θ)
Since y is already expressed as a function of x, we can substitute the value of y from the Cartesian equation:
y = r*sin(θ) = -(√3/3)x
Now, we can substitute the given value of θ (θ=π/6) into the equation to check which option is equivalent to the given equation.
1. Let's substitute θ=π/6 into the equation θ=π/6:
r*sin(π/6) = -(√3/3)*r*cos(π/6)
r*(1/2) = -(√3/3)*r*(√3/2)
1/2 = -3/6
The equation is not satisfied, so θ=π/6 is not the polar equivalent of the given equation.
2. Let's substitute θ=π/3 into the equation:
r*sin(π/3) = -(√3/3)*r*cos(π/3)
r*(√3/2) = -(√3/3)*r*(1/2)
√3/2 = -√3/6
The equation is not satisfied, so θ=π/3 is not the polar equivalent of the given equation.
3. Let's substitute θ=2π/3 into the equation:
r*sin(2π/3) = -(√3/3)*r*cos(2π/3)
r*(√3/2) = -(√3/3)*r*(-1/2)
√3/2 = √3/6
The equation is satisfied, so the polar equation θ=2π/3 is equivalent to the given equation.
4. Let's substitute θ=5π/6 into the equation:
r*sin(5π/6) = -(√3/3)*r*cos(5π/6)
r*(-1/2) = -(√3/3)*r*(-√3/2)
-1/2 = 3/6
The equation is satisfied, so the polar equation θ=5π/6 is also equivalent to the given equation.
Therefore, the polar equivalent to the equation y=−(√3/3)x is θ=2π/3 and θ=5π/6.