How do I find a rectangular-coordinate equation for the curve by elimination the parameter? x=2 sin t, y= 4 cos t
so far I have x= r cos(theta) and y= r sin (theta) but, it's the opposite. please help.
you know that sin^2 + cos^2 = 1, so
(x/2)^2 + (y/4)^2 = 1
x^2/4 + y^2/16 = 1
thank you!!
To eliminate the parameter and derive a rectangular-coordinate equation, you can use the trigonometric identity that links sine and cosine functions:
sin²θ + cos²θ = 1.
Let's start by squaring both equations:
x² = (2 sin t)² = 4 sin²t,
y² = (4 cos t)² = 16 cos²t.
Now, you can substitute sin²t with (1 - cos²t) using the trigonometric identity above:
x² = 4(1 - cos²t) = 4 - 4cos²t.
Next, multiply the x² equation by 4:
4x² = 16 - 16cos²t.
Now, substitute cos²t with (1 - sin²t) using the trigonometric identity, but first, rearrange the equation to isolate cos²t:
4cos²t = 16 - 4x².
Substitute (1 - sin²t) for cos²t:
4(1 - sin²t) = 16 - 4x².
Distribute the 4 and rearrange the equation to have all terms on one side:
4 - 4sin²t = 16 - 4x².
Simplify:
-4sin²t = -4x² + 12.
Divide the entire equation by -4 to solve for sin²t:
sin²t = (1/4)x² - 3.
Finally, substitute sin²t into the y² equation:
y² = 16(1/4)x² - 16(3).
Simplify:
y² = 4x² - 48.
Thus, the rectangular-coordinate equation for the curve is:
y² = 4x² - 48.