what is the cartesian equation of the parametric equations
x=2sin(2t) and y=4cos(2t)
need help badly!!!!!!!
To find the Cartesian equation of parametric equations, we need to eliminate the parameter t by expressing one variable in terms of the other.
Given parametric equations:
x = 2sin(2t)
y = 4cos(2t)
Let's start by squaring both equations:
x^2 = (2sin(2t))^2
y^2 = (4cos(2t))^2
Using the trigonometric identity sin^2θ + cos^2θ = 1, we can rewrite the squared equations as:
x^2 = 4sin^2(2t)
y^2 = 16cos^2(2t)
Since sin^2(2t) = 1 - cos^2(2t), we can substitute this into the first equation:
x^2 = 4(1 - cos^2(2t))
Expanding the equation:
x^2 = 4 - 4cos^2(2t)
Now, let's rearrange the equation to solve for cos^2(2t):
4cos^2(2t) = 4 - x^2
Dividing both sides by 4:
cos^2(2t) = (4 - x^2) / 4
Taking the square root of both sides (note that we choose the positive root since cos^2θ is positive):
cos(2t) = sqrt((4 - x^2) / 4)
Next, we can solve for sin(2t) using the second equation:
y = 4cos(2t)
Substituting the value of cos(2t) we found earlier:
y = 4√((4 - x^2) / 4)
Simplifying the expression:
y = √(4 - x^2)
Therefore, the Cartesian equation of the given parametric equations x = 2sin(2t) and y = 4cos(2t) is:
y = √(4 - x^2)
To find the Cartesian equation of the given parametric equations, we can eliminate the parameter by expressing one variable in terms of the other.
Let's start with the given parametric equations:
x = 2sin(2t)
y = 4cos(2t)
To eliminate t, we can use the trigonometric identity: sin^2(t) + cos^2(t) = 1.
Squaring both of the equations, we get:
x^2 = (2sin(2t))^2 = 4sin^2(2t)
y^2 = (4cos(2t))^2 = 16cos^2(2t)
Now we can use the trigonometric identity to express sin^2(2t) in terms of cos^2(2t):
sin^2(2t) = 1 - cos^2(2t)
Substituting this expression into the equation for x^2, we have:
x^2 = 4(1 - cos^2(2t)) = 4 - 4cos^2(2t)
Now the equation for y^2 becomes:
y^2 = 16cos^2(2t)
Adding these two equations, we obtain the Cartesian equation:
x^2 + y^2 = 4 - 4cos^2(2t) + 16cos^2(2t)
Simplifying:
x^2 + y^2 = 4 + 12cos^2(2t)
Therefore, the Cartesian equation of the given parametric equations x = 2sin(2t) and y = 4cos(2t) is:
x^2 + y^2 = 4 + 12cos^2(2t).
You have to solve each for t, and then equate those results
1) x=2sin(2t)
x/2 = sin(2t)
2t = arcsin(x/2)
t = (1/2)arcsin(x/2)
2)
t = (1/2)arccos(y/4)
so arcsin(x/2) = arccos(y/4)