Given the parametric equations x = 3 + 2sint and y = 2 -4cost

Question

Given the parametric equations x = 3 + 2sint and y = 2 - 4cost for -π/2 ≤ t ≤ π/2

a) find dy/dx in terms of t

b) Find an equation of the tangent line at the point where t = π/4

Answer 1

dx/dt = 2cos(t)

dy/dt = 4sin(t)

a) dy/dx = dy/dt / dx/dt = 4sin(t)/2cos(t) = 2tan(t)

b) dy(π/4)/dx = 2 tan(π/4) = 2 = m

y(π/4) = 2-4cos(π/4) =2-4(√2/2) = 2-2√2

x(π/4) = 3+2(√2/2) = 3+√2

y - (2-2√2) = 2(x-3-√2)

y = 2x -6 -2√2 + 2-2√2

y = 2x-4-4√2