C4 Parametric Differentiation Help!!!

A curve has parametric equations:

x=cos t y=sin 2t

a) Find an expression for dy/dx in terms of the parameter t

b) find the values of the parameter t at the points where dy/dx=0

C) Hence give the exact values of the coordinates of the points on the curve where the tangents are parallel to the x-axis

d) Show that a cartesian equation for the part of the curve where t is greater than and equal to 0 but less than pi is:

y=2x(1-x2)^(1/2)

e) write down a cartesian equation for the part of the curve where t is greater than and equal pi and less than 2 pi.

Hope that is clear it is impossible to write the questions without mathematical symbols. I think the answer to question a is:

dy/dx= 2cos t/-sin t

Any help would be appreciated with working so I can see how you have got the answer! Many thanks.

Reply 1

ykl005

a) y=sin 2t
dy/dt= 2 cos 2t

x=cos t
dx/dt= -sin t

dy/dx = dy/dt * dt/dx
= 2 cos 2t * 1/-sin t
= 2 cos 2t/-sin t

Reply 2

bob_54321

dx/dt = -sint
dy/dt = 2cos 2t
using chain rule dy/dt X dt/dx = dy/dx
dy/dx = -2cos 2t / sin t

the cos definately has to be 2cos 2t, not just 2 cos t.
put that equal to 0

cos 2t = 0
t = pi/4, 3pi/4 (im assuming a range of 0 to pi)
then sub those values in to the original equations to get two pairs of coordinates for c.

Reply 3

bob_54321

for d, if you use sin doulbe angle formula you get y= 2sintcost
y=2sintx
(y/2x)^2 = (sint)^2
x^2= (cost)^2
add them together
y^2/4x^2 + x^2 = 1
y^2=4x^2 - 4x^4
y = 2x(1-x^2)^1/2

Reply 4

bob_54321