the parametric equations of a curve are
x=2theta-sin2theta y=2-cos2theta
show that dy/dx=cot(theta)
find the equation of the tangent to the curve at the point where theta=1/4 pi
for the part of the curve where 0

Question

the parametric equations of a curve are 
x=2theta-sin2theta y=2-cos2theta
show that dy/dx=cot(theta)
find the equation of the tangent to the curve at the point where theta=1/4 pi
for the part of the curve where 0<theta<2pi find the coordinates of the points where the tangent is parallel to the x-axis

abmon98 · Answer

$$dy/dx=dy/d \theta*d \theta/dx$$

abmon98 · Answer

$$2-\cos2\theta=2\sin2\theta $$

anonymous · Answer

$$\frac{dy}{d\theta}=2\sin2\theta\
\frac{dx}{d\theta}=2-2\cos2\theta$$
So
$$\frac{dy}{dx}=\frac{2\sin2\theta}{2-2\cos2\theta}=\frac{\sin2\theta}{1-\cos2\theta}$$
The tangent to the curve when $\theta=\dfrac{\pi}{4}$ will pass through the point $\left(x\left(\dfrac{\pi}{4}\right),y\left(\dfrac{\pi}{4}\right)\right)$ (the point on the curve at which $\theta=\dfrac{\pi}{4}$) and will have a slope of $\dfrac{dy}{dx}\bigg|_{\theta=\pi/4}$ (the derivative evaluated at $\theta=\dfrac{\pi}{4}$).

The tangent line will be parallel to the $x$ axis (i.e. the line $y=0$) when $\dfrac{dy}{dx}=0$, or simply when $\dfrac{dy}{d\theta}=0$. Find the value of $\theta$ between 0 and $2\pi$ that makes this happen.

abmon98 · Answer

Thank you so much :D