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# p3 parametrics watch

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1. part 1:

a curve is given by:

x=1+cos t + sin t
y= 2+ cos t-sin t
t is between 0 and 2pi.

find the gradient of the curve with parameter t (done) and show that the equation of the normal is:

(sin t + cos t)y + (sin t-cos t)x= 3 sin t +cos t

part 2:

by 1st writing in
cos t + sin t=x-1;
cos t-sin t=y-2;

solve the eq. for cos t and sint in terms of x and y.
2. (Original post by innitman_uk)
part 1:

a curve is given by:

x=1+cos t + sin t
y= 2+ cos t-sin t
t is between 0 and 2pi.

find the gradient of the curve with parameter t (done) and show that the equation of the normal is:

(sin t + cos t)y + (sin t-cos t)x= 3 sin t +cos t

part 2:

by 1st writing in
cos t + sin t=x-1;
cos t-sin t=y-2;

solve the eq. for cos t and sint in terms of x and y.
Well for the normal, divide -1 by your gradient to get its gradient. Put that into y = mx + c, use the first 2 equations to find c and it will probably rearrange to give what they want.
3. (Original post by innitman_uk)
part 1:

a curve is given by:

x=1+cos t + sin t
y= 2+ cos t-sin t
t is between 0 and 2pi.

find the gradient of the curve with parameter t (done) and show that the equation of the normal is:

(sin t + cos t)y + (sin t-cos t)x= 3 sin t +cos t
dx/dt = cos t - sin t
dy/dt = -(sin t + cos t)

So dy/dx = dy/dt / dx/dt = (sin t + cos t) / (sin t - cos t).

= -1/(dy/dx)
= (cos t - sin t) / (sin t + cos t).

Equation of normal:

y
= [Gradient of normal]*x + Constant
= [(cos t - sin t) / (sin t + cos t)]*x + Constant
= [(cos t - sin t) / (sin t + cos t)]*(x - (1 + cos t + sin t)) + 2 + cos t - sin t
= [(cos t - sin t) / (sin t + cos t)]*x + 2 + cos t - sin t - [(cos t - sin t) / (sin t + cos t)] - (cos t - sin t)
= [(cos t - sin t) / (sin t + cos t)]*x + 2 - [(cos t - sin t) / (sin t + cos t)]

Multiplying through by (sin t + cos t),

(sin t + cos t)*y
= (cos t - sin t)*x + 2sin t + 2cos t - cos t + sin t
= (cos t - sin t)*x + 3sin t + cos t.

Rearranging,

(sin t - cos t)*x + (sin t + cos t)*y = 3sin t + cos t.

(Original post by innitman_uk)
part 2:

by 1st writing in
cos t + sin t=x-1;
cos t-sin t=y-2;

solve the eq. for cos t and sint in terms of x and y.
x - 1 = cos t + sin t
y - 2 = cos t - sin t

x + y - 3 = 2cos t
=> cos t = x/2 + y/2 - 3/2.

Subtracting,

x - y + 1 = 2sin t
=> sin t = x/2 - y/2 + 1/2.
4. cheers

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