Let be any real number such that .

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

Problem

Let xx be any real number such that 0<x<10 < x < 1. (i) Explain why both arcsin⁡x\arcsin x and arctan⁡x\arctan x lie in the interval (0,π2)\left(0,\tfrac{\pi}{2}\right). (ii) Prove that arcsin⁡x>arctan⁡x\arcsin x > \arctan x for every xx in (0,1)\left(0,1\right). (iii) Determine whether there exists a value of xx in (0,1)\left(0,1\right) for which arcsin⁡x=arctan⁡x\arcsin x = \arctan x. Justify your answer.

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