For , consider the composite expression…

MATHEMATICS · Multi-part · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions

Problem

For θ∈[0,2π]\theta\in[0,2\pi], consider the composite expression arcsin⁡(sin⁡θ)\arcsin(\sin\theta). (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. (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. (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.

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