find the rectangular eqaution of the curve whose parametric equations are x=5cos2T and y=-sin2T, where 0 is less than/equal to T which is less than/equal to 180
from x = 5cos 2T ---> cos 2T = x/5
from y = -sin 2T ---> sin 2T = -y
we know sin^2 2T + cos^2 2T= 1
x^2 /25 + y^2 = 1
x^2 + 25y^2 = 25
(looks like we have an ellipse)
test:
let T = 30°
x = 5cos60° = 5(1/2) = 5/2
y = -sin60° = -√3/2
sub into the rectangular...
LS = 25/4 + 25(3/4) = 100/4 = 25
let T = 19.6°
x = 5cos39.2· = appr. 3.8747
y = -sin39.2 = appr. -.6320
LS = 3.8747^2 + 25(.6320)^2 = 24.99999
looks promising.
Why did the parametric equations go to the circus? Because they wanted to be rectangular! Let's turn their show into a rectangle.
To find the rectangular equation of the curve, we need to eliminate the parameter T. Let's use the cosine^2(T) + sine^2(T) = 1 identity and rearrange the parametric equations:
x = 5cos(2T)
= 5cos^2(T) - 5sin^2(T)
y = -sin(2T)
= -2sin(T)cos(T)
Now, let's square both sides of these equations to eliminate the cos(T) and sin(T):
x^2 = 25cos^4(T) - 50cos^2(T)sin^2(T) + 25sin^4(T)
y^2 = 4sin^2(T)cos^2(T)
By subtracting the two equations, we can eliminate the sin(T) terms:
x^2 - y^2 = 25cos^4(T) - 50cos^2(T)sin^2(T) + 25sin^4(T) - 4sin^2(T)cos^2(T)
Simplifying this equation, we get:
x^2 - y^2 = 25cos^4(T) - 50cos^2(T)sin^2(T) + 25sin^4(T) - 4sin^2(T)cos^2(T)
= 25(cos^4(T) - sin^4(T)) - 2sin^2(T)cos^2(T)
Using the identity cos^2(T) - sin^2(T) = cos(2T), we can simplify further:
x^2 - y^2 = 25cos(2T) - 2sin^2(T)cos^2(T)
= 25cos(2T) - 2(1 - cos^2(T))cos^2(T)
= 25cos(2T) - 2cos^2(T) + 2cos^4(T)
And voila! The rectangular equation of the curve is:
x^2 - y^2 = 25cos(2T) - 2cos^2(T) + 2cos^4(T)
Now that our parametric equations have been transformed into a rectangular equation, they're finally ready for their rectangular show!
To find the rectangular equation of the curve defined by the parametric equations x = 5cos(2T) and y = -sin(2T), we can eliminate the parameter T by expressing it in terms of x and y.
Let's start with the equation for x:
x = 5cos(2T)
We can rewrite this equation as:
cos(2T) = x/5
Now, let's work with the equation for y:
y = -sin(2T)
We can rewrite this equation as:
sin(2T) = -y
Next, we'll use the trigonometric identity sin^2(θ) + cos^2(θ) = 1 to eliminate the parameter T. Since our equations involve sin(2T) and cos(2T), we'll use the double-angle formula:
sin^2(θ) + cos^2(θ) = 1
2sin^2(θ) + 2cos^2(θ) = 2
Now, we can rewrite our trigonometric identities as follows:
sin^2(2T) = (1 - cos^2(2T))
cos^2(2T) = (1 - sin^2(2T))
Substituting the equations for sin(2T) and cos(2T), we get:
sin^2(2T) = (1 - (x/5)^2)
cos^2(2T) = (1 - y^2)
Using these equations, we can now obtain the rectangular equation:
cos^2(2T) - sin^2(2T) = (1 - y^2) - (1 - (x/5)^2)
cos^2(2T) - sin^2(2T) = (1 - y^2) - 1 + (x/5)^2
cos^2(2T) - sin^2(2T) = (x/5)^2 - y^2
Now, using the trigonometric identity cos^2(θ) - sin^2(θ) = cos(2θ), we can simplify this equation:
cos(2T) = (x/5)^2 - y^2
So, the rectangular equation of the curve defined by the parametric equations x = 5cos(2T) and y = -sin(2T) is:
cos(2T) = (x/5)^2 - y^2
To find the rectangular equation of a parametric curve, we need to eliminate the parameter (T) and express x and y solely in terms of each other.
Given the parametric equations:
x = 5cos(2T)
y = -sin(2T)
We can use trigonometric identities to eliminate T and express x and y in terms of each other.
Let's start with the equation for x:
x = 5cos(2T)
Using the double angle formula for cosine, we have:
x = 5[cos^2(T) - sin^2(T)]
Next, apply the Pythagorean identity cos^2(T) + sin^2(T) = 1:
x = 5[1 - sin^2(T) - sin^2(T)]
x = 5 - 10sin^2(T)
Now, let's work with the equation for y:
y = -sin(2T)
Using the double angle formula for sine, we have:
y = -2sin(T)cos(T)
Using the Pythagorean identity cos^2(T) + sin^2(T) = 1:
y = -2sin(T)sqrt(1 - sin^2(T))
Therefore, the rectangular equation of the curve is:
x = 5 - 10sin^2(T)
y = -2sin(T)sqrt(1 - sin^2(T))
Note: Keep in mind that the given parameter range is 0 ≤ T ≤ 180.