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 to and opposite θ. suppose also that x and y vary with time. at a certain instant x=6 units and is increasing at 3 unit/s, while y=9 and is decreasing at 1/7 units/s. how fast is θ changing at that instant?

Answer 1

y/x = tan(θ)
θ = arctan(y/x)
θ' = (y'/x - y*x'/x^2)/(1 + y^2/x^2)
θ' = ((-1/9)/5 - 1*6/5^2)/(1 + 1^2/5^2) ... substitute given values
 .. = (-1/45 - 6/25)/(1 + 1/25)
... simplify a bit .. = (-25/45 - 6)/26
 ... multiply numerator and denominator by 25
  .. = -295/(45*26) = -59/234 ... radians per second