(a) Prove that for every . (b) Using the result of part (a), evaluate the following,…

MATHEMATICS · Subjective · CBSE · Mathematics · Inverse Trigonometric Functions · Inverse Trigonometric Functions

Problem

(a) Prove that arcsin⁡(−x)=−arcsin⁡x\arcsin(-x)=-\arcsin x for every x∈[−1,1]x\in[-1,1]. (b) Using the result of part (a), evaluate the following, giving each answer as a principal value: (i) arcsin⁡(sin⁡(−11π6))\arcsin\bigl(\sin\bigl(-\dfrac{11\pi}{6}\bigr)\bigr) (ii) arcsin⁡(sin⁡(−5π4))\arcsin\bigl(\sin\bigl(-\dfrac{5\pi}{4}\bigr)\bigr) (c) Explain briefly why the answers obtained in part (b) lie in the principal value range of arcsin⁡x\arcsin x.

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