Question

Let θ (in radians) be an acute angle in a right triangle, and let x and y, respectively, be the lengths of the sides adjacent and opposite θ. Suppose also that x and y vary with time.

a. How are dθ/dt, dx/dt and dy/dt related?

Please give steps and explain!

Answer 1

dθ/dt = [(cos^2 θ)*(dy/dt * x - y * dx/dt)]/(x^2)

Explanation:

Given that x and y are the lengths of the sides adjacent and opposite θ, then they are related by:

tan θ = y/x

Differentiating respect to t, we get:

sec^2 θ * dθ/dt = (dy/dt * x - y * dx/dt)/(x^2)

dθ/dt = [(cos^2 θ)*(dy/dt * x - y * dx/dt)]/(x^2)

Answer 2

Explanation is given step by step just below:
tan(theta(t))=y(t)/x(t)

differentiate
sec^2(theta(t))*theta'(t)=y'(t)x(t)-y(t)x'(t)/x^2(t)
thus
theta'(t)=(y'(t)x(t)-y(t)x'(t))
divided by (x^2(t)*sec^2(θ(t))