Given the parametric equations below, eliminate the parameter t to obtain a Cartesian equation.
0
≤
t
≤
2
π
{
x
(
t
)
=
8
sin
(
t
)
y
(
t
)
=
6
cos
(
t
)
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To eliminate the parameter t and obtain a Cartesian equation, we need to express x and y in terms of each other.
Given:
x(t) = 8*sin(t)
y(t) = 6*cos(t)
To eliminate t, we can use the trigonometric identity: sin^2(t) + cos^2(t) = 1
Squaring both equations:
x^2(t) = (8*sin(t))^2 = 64*sin^2(t)
y^2(t) = (6*cos(t))^2 = 36*cos^2(t)
Dividing the equation for x^2(t) by 64 and the equation for y^2(t) by 36:
x^2(t)/64 = sin^2(t)
y^2(t)/36 = cos^2(t)
Since sin^2(t) + cos^2(t) = 1, we can substitute these equations into the identity:
x^2(t)/64 + y^2(t)/36 = 1
Therefore, the Cartesian equation is:
(x^2/64) + (y^2/36) = 1
To eliminate the parameter t and obtain a Cartesian equation, we need to express x and y solely in terms of each other without any reference to t.
Using the trigonometric identity sin^2(t) + cos^2(t) = 1, we can square both equations and simplify:
x(t) = 8sin(t)
y(t) = 6cos(t)
x^2(t) = (8sin(t))^2 = 64sin^2(t)
y^2(t) = (6cos(t))^2 = 36cos^2(t)
We can then use the identity sin^2(t) + cos^2(t) = 1 to rewrite the equations:
x^2(t) = 64(1 - cos^2(t))
y^2(t) = 36cos^2(t)
Next, divide both sides of the equations by 64 and 36 respectively:
x^2(t)/64 = 1 - cos^2(t)
y^2(t)/36 = cos^2(t)
Rearranging the equations:
1 - cos^2(t) = x^2(t)/64
cos^2(t) = y^2(t)/36
Now, substitute 1 - cos^2(t) with sin^2(t) using the trigonometric identity:
sin^2(t) = x^2(t)/64
cos^2(t) = y^2(t)/36
Finally, taking the square root of both equations:
sin(t) = x(t)/8
cos(t) = y(t)/6
Since sin(t) = x(t)/8 and cos(t) = y(t)/6, we can substitute these values into the trigonometric equation sin^2(t) + cos^2(t) = 1:
(x(t)/8)^2 + (y(t)/6)^2 = 1
This is the Cartesian equation that eliminates the parameter t.