Eliminate the parameter to find a description of the following circles or circular arc in terms of x and y. Give the center and radius and indicate the positive orientation.
x=3sin(t), y=-3cos(t) ; 0 < t < 2pi
x=3sin(t), y=-3cos(t) ; 0 < t < 2pi
x^2 = 9 sin^2 t
y^2 = 9 cos^2 t
x^2 + y^2 = 9 (sin^2 t + cos^2 t)
but we all know sin^2+cos^2 = 1
x^2 + y^2 = 3^2
center at (0,0) radius 3
x=3sin(t), y=-3cos(t)
sint = x/3
cost = y/-3
sin^2 t + cos^2 t = 1
x^2/9 + y^2/9 = 1
x^2 + y^2 = 9
circle with centre (0,0) and radius 3
verification:
https://www.wolframalpha.com/input/?i=parametric+plot+x%3D3sin%28t%29%2C+y%3D-3cos%28t%29+
To eliminate the parameter t and express the equation in terms of x and y, we can use the trigonometric identity sin^2(t) + cos^2(t) = 1.
Given x = 3sin(t) and y = -3cos(t), we can square both equations to obtain:
x^2 = (3sin(t))^2 = 9sin^2(t)
y^2 = (-3cos(t))^2 = 9cos^2(t)
Now, substituting sin^2(t) = 1 - cos^2(t) into the equation for x^2, we have:
x^2 = 9(1 - cos^2(t))
Next, we can rewrite cos^2(t) as 1 - sin^2(t):
x^2 = 9(1 - (1 - sin^2(t)))
= 9sin^2(t)
Since we know that sin^2(t) + cos^2(t) = 1, we can simplify further:
x^2 = 9(1 - cos^2(t))
= 9cos^2(t)
From the equations x^2 = 9sin^2(t) and y^2 = 9cos^2(t), we can see that x^2 + y^2 = 9(sin^2(t) + cos^2(t)) = 9.
This shows that the equation x^2 + y^2 = 9 represents a circle with center (0, 0) and radius 3. The positive orientation indicates that the circle is traced counterclockwise as t increases from 0 to 2π.