Question

Suppose the point (pi/3, pi/4) is on the curve sinx/x + siny/y = C, where C is a constant. Use the tangent line approximation to find the y-coordinate of the point on the curve with x-coordinate pi/3 + pi/180.

Answer 1

Using implicit derivatives,
y' =
y^2(sinx - x cosx)
------------------------------
x^2(y cosy - siny)

At (π/3,π/4), that is y' = 3(3√3-π) / 4√2(π-4) ≈ 1.27

Now, using the approximation ∆y/∆x ≈ dy/dx we get
y1 = y0 + y' (x1-x0)
= π/4 + 1.27(π/180) ≈ 0.81

Answer 2

Just an edit to the previous response, 3(3√3-π) / 4√2(π-4) is NOT equal to 1.27, but rather about -1.27

Answer 3

This will also lead y(1) from the tangent line approximation to be equal to approximately .76323248, NOT .81