Find a rectangular equation for the plane curve defined by the parametric equations.
x = sin(theta)
y = 3cos(theta)
we know sinØ = y/r and cosØ = x/r
and tanØ = y/x
then for the given:
sinØ = x = y/r
rx = y ---> r = y/x
3cosØ = y = 3x/r
ry = 3x
(y/x)y = 3x
y^2 = 3x^2
y = ±√3 x
To find the rectangular equation for the given parametric equations, we need to eliminate the parameter (θ) and express the relationship between x and y in terms of a single equation.
From the given parametric equations:
x = sin(θ)
y = 3cos(θ)
We'll use the trigonometric identity: sin^2(θ) + cos^2(θ) = 1.
Square both equations:
x^2 = sin^2(θ)
y^2 = 9cos^2(θ)
Divide the second equation by 9:
y^2/9 = cos^2(θ)
Now, substitute this value for cos^2(θ) into the first equation:
x^2 = sin^2(θ) = 1 - cos^2(θ)
Rearranging the equation:
x^2 + cos^2(θ) = 1
Using the trigonometric identity, we substitute cos^2(θ) for y^2/9:
x^2 + y^2/9 = 1
Finally, multiplying both sides by 9:
9x^2 + y^2 = 9
Therefore, the rectangular equation for the plane curve defined by the parametric equations x = sin(θ) and y = 3cos(θ) is 9x^2 + y^2 = 9.
To find a rectangular equation for the plane curve defined by the parametric equations, we need to eliminate the parameter theta and express x and y in terms of each other.
Given the parametric equations:
x = sin(theta)
y = 3cos(theta)
We can use a trigonometric identity to eliminate theta:
From the equation x = sin(theta), we can rearrange it as:
sin(theta) = x
Similarly, from the equation y = 3cos(theta), we can rearrange it as:
cos(theta) = y/3
Now, since sin^2(theta) + cos^2(theta) = 1, we can replace sin^2(theta) with (1 - cos^2(theta)) in the equation sin(theta) = x:
(1 - cos^2(theta)) = x^2
1 - cos^2(theta) = x^2
Next, we can substitute cos^2(theta) with (1 - sin^2(theta)):
1 - (1 - sin^2(theta)) = x^2
sin^2(theta) = x^2
Finally, we can take the square root of both sides to get the rectangular equation:
sin(theta) = ± √x^2
sin(theta) = ± x
Therefore, the rectangular equation for the plane curve would be:
y = ± 3√(1 - x^2), where y can take positive or negative values.