Mathematics

Mathematics

Problem

For θ∈[0,2π]\theta\in[0,2\pi], consider the composite expression arcsin⁡(sin⁡θ)\arcsin(\sin\theta). a. (i) Determine the value of arcsin⁡(sin⁡θ)\arcsin(\sin\theta) when θ∈[0,π2]\theta\in\bigl[0,\tfrac{\pi}{2}\bigr]. Justify your answer with reference to the principal value branch. b. (ii) Calculate arcsin⁡(sin⁡7π6)\arcsin(\sin\tfrac{7\pi}{6}) and arcsin⁡(sin⁡5π4)\arcsin(\sin\tfrac{5\pi}{4}) showing each step clearly. c. (iii) Derive a piece-wise definition of arcsin⁡(sin⁡θ)\arcsin(\sin\theta) for all θ\theta in the interval [0,2π][0,2\pi]. Provide clear reasoning for the interval break-up used.

Browse the problem library