Question

The parametric equations of a curve are x = 2theta + sin 2 theta, y=1-cos 2 theta. Show that dy/dx = tan theta.

Answer 1

dy / dx = {dy/dt} / {dx/dt}

Answer 2

dt is dtheta?

Answer 3

yep

Answer 4

dx/dt would be 2-2cos2theta?

Answer 5

if x = theta + sin 2 theta as per question stated above then:
dx/dtheta = 1 + 2 cos 2 theta

Answer 6

oh oops, supposed to be 2theta + sin 2 theta. ive corrected. so dx/dt would be 2 + 2cos2theta?

Answer 7

y

Answer 8

?

Answer 9

yes

Answer 10

Dy/dt = 2sin2theta right?

Answer 11

@perl helppp

Answer 12

Chain rule.

Answer 13

$$
\Large \frac{dy}{dx }=\frac{\frac{dy}{d\theta}}{\frac {dx}{d\theta}}= \frac{2sin(2\theta)}{1 + 2 cos 2 \theta}




$$

Answer 14

but that doesnt equal tan theta.
i think i copied your question wrong

Answer 15

@perl oh it's 2sin2theta / 2 + 2 cos 2 theta

Answer 16

how did you get 2 + 2 cos 2 theta for the denominator?
x = theta + sin(2 theta)

Answer 17

x = theta + sin(2 theta)
dx/dtheta = 1 + 2 cos(2 theta)

can you double check your directions , did I copy correctly

Answer 18

I've edited the ques, supposedly 2theta + sin2theta.

Answer 19

ok i see now :)

Answer 20

Idk how 2sin2theta / 2+2cos2theta = tan theta ):

Answer 21

i am using two identities
sin (2u) = 2 sin u cos u
cos(2u) = 2cos^2(u) - 1

Answer 22

Uh huh

Answer 23

$$
\Large{ \frac{dy}{dx }=\frac{\frac{dy}{d\theta}}{\frac {dx}{d\theta}}= \frac{2\sin(2\theta)}{2 + 2 \cos 2 \theta}\\= \frac{2\sin(2\theta)}{2(1 + \cos 2 \theta) }=\frac{\sin(2\theta)}{1 + \cos (2 \theta) }\\
= \frac{\sin(2\theta)}{1 + \cos(2\theta) }
=\frac{2\sin(\theta)\cos(\theta)}{1 + (2\cos^2\theta-1) }
=\frac{2\sin(\theta)\cos(\theta)}{2\cos^2\theta }\\
=\frac{\sin(\theta)}{\cos (\theta) }= \tan \theta
\\

\\

}

$$

Answer 24

Thanksss