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Question Number 63747 by Tony Lin last updated on 08/Jul/19

What does sin^(−2) x mean?

$${What}\:{does}\:{sin}^{−\mathrm{2}} {x}\:{mean}? \\ $$

Commented by MJS last updated on 08/Jul/19

that′s the question... we′re not consequent in these things f^(−1) (x) means the inverse function of f(x) some people use f^1 (x)=f′(x) f^2 (x)=f′′(x) ... for trigonometric functions usually sin^(−1) x =arcsin x cos^(−1) x =arccos x ... but sin^2 x =(sin x)^2 ... so sin^(−2) x could mean different things (1/(sin^2 x)) arcsin^2 x (1/((sin x)′′)) it′s a shame

$$\mathrm{that}'\mathrm{s}\:\mathrm{the}\:\mathrm{question}... \\ $$$$\mathrm{we}'\mathrm{re}\:\mathrm{not}\:\mathrm{consequent}\:\mathrm{in}\:\mathrm{these}\:\mathrm{things} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)\:\mathrm{means}\:\mathrm{the}\:\mathrm{inverse}\:\mathrm{function}\:\mathrm{of}\:{f}\left({x}\right) \\ $$$$\mathrm{some}\:\mathrm{people}\:\mathrm{use} \\ $$$${f}^{\mathrm{1}} \left({x}\right)={f}'\left({x}\right) \\ $$$${f}^{\mathrm{2}} \left({x}\right)={f}''\left({x}\right) \\ $$$$... \\ $$$$\mathrm{for}\:\mathrm{trigonometric}\:\mathrm{functions}\:\mathrm{usually} \\ $$$$\mathrm{sin}^{−\mathrm{1}} \:{x}\:=\mathrm{arcsin}\:{x} \\ $$$$\mathrm{cos}^{−\mathrm{1}} \:{x}\:=\mathrm{arccos}\:{x} \\ $$$$... \\ $$$$\mathrm{but} \\ $$$$\mathrm{sin}^{\mathrm{2}} \:{x}\:=\left(\mathrm{sin}\:{x}\right)^{\mathrm{2}} \\ $$$$... \\ $$$$\mathrm{so} \\ $$$$\mathrm{sin}^{−\mathrm{2}} \:{x}\:\mathrm{could}\:\mathrm{mean}\:\mathrm{different}\:\mathrm{things} \\ $$$$\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \:{x}} \\ $$$$\mathrm{arcsin}^{\mathrm{2}} \:{x} \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{sin}\:{x}\right)''} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{a}\:\mathrm{shame} \\ $$

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