Question

One side of a right triangle is known to be 36 cm long and the opposite angle is measured as 30°, with a possible error of ±1°. (a) Use differentials to estimate the error in computing the length of the hypotenuse. (Round your answer to two decimal places.)

Answer 1

Answer:

  ±2.18 cm

Step-by-step explanation:

The definition of the sine of an angle (θ) is ...

  Sin = Opposite/Hypotenuse

Then the hypotenuse (h) is found from ...

  h = Opposite/sin(θ)

Differentiating with respect to θ gives ...

  dh/dθ = Opposite×(-cos(θ)/sin²(θ))

For the given numbers, this is ...

  dh/dθ = (36 cm)×(-√3/2)/(1/2)² = -72√3 cm/radian

The error of ±1° corresponds to ±π/180 radians, so the error is estimated to be ...

  error = (dh/dθ)Δθ = (-72√3 cm/radian)(±π/180 radians) = ±0.4π√3 cm

  error = ±2.18 cm