SOLUTION: Write as a single equation in x and y (a) {x=t+1, y=t^2-t} (b) {x= 3sqrt(t-1), y=t^2-t} (c) {x=sin(t), y=cos(t)}

Algebra ->  Trigonometry-basics -> SOLUTION: Write as a single equation in x and y (a) {x=t+1, y=t^2-t} (b) {x= 3sqrt(t-1), y=t^2-t} (c) {x=sin(t), y=cos(t)}      Log On


   

Question 1157184: Write as a single equation in x and y
(a) {x=t+1, y=t^2-t}
(b) {x= 3sqrt(t-1), y=t^2-t}
(c) {x=sin(t), y=cos(t)}
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Write as a single equation in x and y
(a)
 
matrix%281%2C5%2C++%22%7B%22%2Cx=t%2B1%2C%22%2C%22%2Cy=t%5E2-t%2C%22%7D%22%29

Factor t out of the right side of the second equation"

matrix%281%2C5%2C++%22%7B%22%2Cx=t%2B1%2C%22%2C%22%2Cy=t%28t-1%29%2C%22%7D%22%29

Solve the first equation for t:
x=t%2B1
x-1=t
Substitute x-1 for t in the second equation:
    
leave it like that or FOIL out the right side:
y=x%5E2-3x%2B2
(b) {x= 3root t-1, y=t^2-t}
Factor t out of the right side of the second equation"

matrix%281%2C5%2C++%22%7B%22%2Cx=3%2Asqrt%28t-1%29%2C%22%2C%22%2Cy=t%28t-1%29%2C%22%7D%22%29
Square both sides of the first equation:
x%5E2=%283%2Asqrt%28t-1%29%29%5E2
x%5E2=9%2A%28t-1%29
x%5E2=9t-9
x%5E2%2B9=9t
%28x%5E2%2B9%29%2F9=t
Substitute (x²+9)/9 for t in the second equation:
y=%28%28x%5E2%2B9%29%2F9%5E%22%22%29%28%28%28x%5E2%2B9%29%2F9%5E%22%22%29%5E%22%22-1%29

y=%28%28x%5E2%2B9%29%2F9%5E%22%22%29%28%28x%5E2%2B9-9%29%2F9%5E%22%22%29
y=%28%28x%5E2%2B9%29%2F9%5E%22%22%29%28x%5E2%2F9%5E%22%22%29
y+=+x%5E2%28x%5E2+%2B+9%29%2F81%5E%22%22
(c) {x=sin t , y=cos t}
This one's different. Use the trig identity sin²(θ)+cos²(θ)=1.

sin%5E2%28t%29%2Bcos%5E2%28t%29=1
x%5E2%2By%5E2=1
Leave it like that or solve for y.

Edwin