How to do that?

Question

The parametric equations for the motion of a charged particle released from rest in electric and magnetic fields at right angles to each other take the forms x=a(θ−sinθ),y=a(1−cosθ). Show that the tangent to the curve has slope cot(θ/2). Use this result at a few calculated values of x and y to sketch the form of the particle's trajectory.

Answer 1

We need to calculate dy/dx, which is the slope of any tangent.

By the chain rule

dy/dθ = dy/dx dx/dθ

Thus

dy/dx = (dy/dθ) / (dx/dθ)

= (asinθ) / (a(1-cosθ))

= sinθ / (1-cosθ)

= cot(θ/2)

If you need help with the last identity, let me derive it using the following

formulas

sin(2u) = 2sin(u)cos(u)

cos(2u) = 1 - 2sin²(u)

Let u = θ/2

Then

sinθ / (1-cosθ) = sin(2u) / (1-cos(2u))

= 2sin(u)cos(u) / 2sin²(u)

= cos(u) / sin(u)

= cot(u)

= cot(θ/2)