What are parametric equations?
How would you find the rectangular eq's when given parametric eq's? and how would you find parametric eq's when given rectangular eq's?
for example,
given x=cos4t and y=sin4t (t is between 0 and 2pi) what is the retangular equation?
given y=2x^(2)+4 whatare the parametric equations?
Parametric equations are a set of equations that represent the relationship between two variables (usually denoted as x and y) in terms of a third variable (usually denoted as t). In parametric equations, each variable is expressed as a function of the parameter t.
To find the rectangular equations when given parametric equations, you need to eliminate the parameter t and express x and y solely in terms of each other. Here's how you can do it:
1. Given the parametric equations:
x = cos(4t)
y = sin(4t)
2. Use a trigonometric identity, such as cos²(t) + sin²(t) = 1, to eliminate the parameter t. Dividing the two equations, we get:
(x / cos(4t)) = (y / sin(4t))
3. Simplify the equation by multiplying both sides by the denominators:
xsin(4t) = ycos(4t)
4. Apply trigonometric identities (sin(2u) = 2sin(u)cos(u)) to further simplify the equation:
2xsin(2t) = 2ycos(2t)
5. Finally, simplify by using double-angle identities (sin(2u) = 2sin(u)cos(u), cos(2u) = cos²(u) - sin²(u)) to write the equation in terms of x and y:
2xsin(2t) = 2ycos(2t)
2x(2sin(t)cos(t)) = 2y(cos²(t) - sin²(t))
4xsin(t)cos(t) = 2y(cos²(t) - sin²(t))
2xsin(2t) = y(cos(2t))
Therefore, the rectangular equation is:
2xsin(2t) = y(cos(2t))
To find the parametric equations when given rectangular equations, you need to solve for x and y in terms of t. Here's how you can do it:
1. Given the rectangular equation:
y = 2x² + 4
2. Assume x is a function of t (x = f(t)), and substitute it into the equation:
y = 2(f(t))² + 4
3. Solve for f(t):
2(f(t))² + 4 = y
(f(t))² = (y - 4) / 2
f(t) = √((y - 4) / 2)
4. Therefore, the parametric equations would be:
x = √((y - 4) / 2)
y = 2x² + 4
In the given example where y = 2x² + 4, the parametric equations would be:
x = √((y - 4) / 2)
y = 2x² + 4