If dy/dx = sin x / cos y and y (0) = 3pi/2, find an equation for y in terms of x

dy / dx = sin x / cos y

We rewrite the equation:

(cos (y) * dy) = (sin (x) * dx)

We integrate both sides of the equation:

sin (y) = - cos (x) + C

We use the initial condition to find the constant C:

sin (3pi / 2) = - cos (0) + C

-1 = - 1 + C

C = - 1 + 1

C = 0

The equation is then:

sin (y) = - cos (x)

Clearing y:

y = Arcosine (-cos (x))

Answer:

An equation for and in terms of x is:

y = Arcosine (-cos (x))