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Question Number 56200 by naka3546 last updated on 11/Mar/19

Commented by Abdo msup. last updated on 12/Mar/19

changement x=sinθ give I =∫_0 ^(π/2) ((arcos(cosθ)cosθ)/(cosθ)) dθ =∫_0 ^(π/2) θ dθ =[(θ^2 /2)]_0 ^(π/2) =(π^2 /8) .

$${changement}\:{x}={sin}\theta\:{give}\: \\ $$$${I}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{arcos}\left({cos}\theta\right){cos}\theta}{{cos}\theta}\:{d}\theta\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \theta\:{d}\theta \\ $$$$=\left[\frac{\theta^{\mathrm{2}} }{\mathrm{2}}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:=\frac{\pi^{\mathrm{2}} }{\mathrm{8}}\:. \\ $$

Commented by naka3546 last updated on 12/Mar/19

Commented by naka3546 last updated on 12/Mar/19

how about this ? Is it same as before ?

$${how}\:\:{about}\:\:{this}\:?\:{Is}\:{it}\:\:{same}\:\:{as}\:\:{before}\:? \\ $$

Commented by MJS last updated on 12/Mar/19

usually cos^n x means (cos x)^n . somehow it′s only used witb n≥1. cos^(−1) x means arccos x. this is the same principle as y=f(x), x=f^(−1) (y). but here many people write f′(x) for the 1^(st) derivate and f^n (x) for the n^(th) derivate... once there had been rules but with the internet confusion came...

$$\mathrm{usually}\:\mathrm{cos}^{{n}} \:{x}\:\mathrm{means}\:\left(\mathrm{cos}\:{x}\right)^{{n}} .\:\mathrm{somehow} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{only}\:\mathrm{used}\:\mathrm{witb}\:{n}\geqslant\mathrm{1}.\:\mathrm{cos}^{−\mathrm{1}} \:{x}\:\mathrm{means} \\ $$$$\mathrm{arccos}\:{x}.\:\mathrm{this}\:\mathrm{is}\:\mathrm{the}\:\mathrm{same}\:\mathrm{principle}\:\mathrm{as} \\ $$$${y}={f}\left({x}\right),\:{x}={f}^{−\mathrm{1}} \left({y}\right).\:\mathrm{but}\:\mathrm{here}\:\mathrm{many}\:\mathrm{people} \\ $$$$\mathrm{write}\:{f}'\left({x}\right)\:\mathrm{for}\:\mathrm{the}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{derivate}\:\mathrm{and}\:{f}^{{n}} \left({x}\right) \\ $$$$\mathrm{for}\:\mathrm{the}\:{n}^{\mathrm{th}} \:\mathrm{derivate}...\:\mathrm{once}\:\mathrm{there}\:\mathrm{had}\:\mathrm{been} \\ $$$$\mathrm{rules}\:\mathrm{but}\:\mathrm{with}\:\mathrm{the}\:\mathrm{internet}\:\mathrm{confusion}\:\mathrm{came}... \\ $$

Commented by Kunal12588 last updated on 12/Mar/19

i will glad to know those rules I heard this from our teacher− Cosine(any trig func) should start with lowercase letter only ′cos x′ is right but ′Cos′ is wrong. nowadays internet doesn′t follow this rule. Also he says (√3) was 1.732 on his days then 1.73 in next generation in our this is 1.7 then our next would say (√3)=1

$${i}\:{will}\:{glad}\:{to}\:{know}\:{those}\:{rules} \\ $$$${I}\:{heard}\:{this}\:{from}\:{our}\:{teacher}− \\ $$$${Cosine}\left({any}\:{trig}\:{func}\right)\:{should}\:{start}\:{with} \\ $$$${lowercase}\:{letter}\:{only}\:'{cos}\:{x}'\:{is}\:{right}\:{but}\:'{Cos}' \\ $$$${is}\:{wrong}.\:{nowadays}\:{internet}\:{doesn}'{t}\:{follow} \\ $$$${this}\:{rule}.\:{Also}\:{he}\:{says}\:\sqrt{\mathrm{3}}\:{was}\:\mathrm{1}.\mathrm{732}\:{on}\:{his} \\ $$$${days}\:{then}\:\mathrm{1}.\mathrm{73}\:{in}\:{next}\:{generation}\:{in}\:{our} \\ $$$${this}\:{is}\:\mathrm{1}.\mathrm{7}\:{then}\:{our}\:{next}\:{would}\:{say}\:\sqrt{\mathrm{3}}=\mathrm{1} \\ $$

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