known-a-lt-pi-2-If-M-lt-1-with-cos-x-cos-y-M-x-y-for-every-x-y-0-a-then-M-

Question Number 55908 by gunawan last updated on 06/Mar/19

$$\mathrm{known}\:{a}\:<\:\frac{\pi}{\mathrm{2}}\:. \\ $$$$\mathrm{If}\:\:\mathrm{M}<\mathrm{1}\:\mathrm{with}\:\mid\mathrm{cos}\:{x}−\mathrm{cos}\:{y}\mid\leqslant\mathrm{M}\:\mid{x}−{y}\mid \\ $$$$\mathrm{for}\:\mathrm{every}\:{x},\:{y}\:\in\:\left[\mathrm{0},{a}\right],\:\mathrm{then}\:\mathrm{M}=.. \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 07/Mar/19

$${let}\:{y}>{x} \\ $$$${so}\:{cosx}>{cosy} \\ $$$$={cosx}−{cosy} \\ $$$$=+{ve} \\ $$$${but}\:{x}−{y}=−{ve} \\ $$$$\mid{x}−{y}\mid=+{ve} \\ $$$${from}\:{attached}\:{graph} \\ $$$${tan}\theta=\frac{{y}_{\mathrm{2}} −{y}_{\mathrm{1}} }{{x}_{\mathrm{2}} −{x}_{\mathrm{1}} }=\frac{{cosy}−{cosx}}{{y}−{x}} \\ $$$${tan}\theta={M}=\frac{{cosy}−{cosx}}{{y}−{x}}\:{but}\:{cosx}>{cosy} \\ $$$${y}>{x}\:\:{so}\:{tan}\theta={M}\:\:{is}\:−{ve} \\ $$$${wait}… \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 07/Mar/19

Commented by gunawan last updated on 07/Mar/19

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:+{ve}\:\mathrm{Sir}? \\ $$$$ \\ $$

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