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Question Number 13401 by Tinkutara last updated on 19/May/17

If sin (π cos θ) = cos (π sin θ), then prove that sin 2θ = ± (3/4)

$$\mathrm{If}\:\mathrm{sin}\:\left(\pi\:\mathrm{cos}\:\theta\right)\:=\:\mathrm{cos}\:\left(\pi\:\mathrm{sin}\:\theta\right),\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{sin}\:\mathrm{2}\theta\:=\:\pm\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Answered by ajfour last updated on 19/May/17

sin x=cos y x=(π/2)±y (at least) here sin (πcos θ)=cos (πsin θ) πcos θ=(π/2)+πsin θ ......(i) πcos θ=(π/2)−πsin θ ......(ii) from (i): cos θ−sin θ=(1/2) , squaring 1−sin 2θ=(1/4) ⇒ sin 2θ=(3/4) ; from (ii): cos θ+sin θ=(1/2) , squaring 1+sin 2θ=(1/4) ⇒ sin 2θ=−(3/4) .

$$\mathrm{sin}\:{x}=\mathrm{cos}\:{y} \\ $$$${x}=\frac{\pi}{\mathrm{2}}\pm{y}\:\:\:\:\:\left({at}\:{least}\right) \\ $$$${here}\:\:\:\:\mathrm{sin}\:\left(\pi\mathrm{cos}\:\theta\right)=\mathrm{cos}\:\left(\pi\mathrm{sin}\:\theta\right) \\ $$$$\pi\mathrm{cos}\:\theta=\frac{\pi}{\mathrm{2}}+\pi\mathrm{sin}\:\theta\:\:\:\:......\left({i}\right) \\ $$$$\pi\mathrm{cos}\:\theta=\frac{\pi}{\mathrm{2}}−\pi\mathrm{sin}\:\theta\:\:\:......\left({ii}\right) \\ $$$${from}\:\left({i}\right): \\ $$$$\mathrm{cos}\:\theta−\mathrm{sin}\:\theta=\frac{\mathrm{1}}{\mathrm{2}}\:\:\:,\:{squaring} \\ $$$$\mathrm{1}−\mathrm{sin}\:\mathrm{2}\theta=\frac{\mathrm{1}}{\mathrm{4}}\:\:\Rightarrow\:\:\:\mathrm{sin}\:\mathrm{2}\theta=\frac{\mathrm{3}}{\mathrm{4}}\:; \\ $$$${from}\:\left({ii}\right): \\ $$$$\mathrm{cos}\:\theta+\mathrm{sin}\:\theta=\frac{\mathrm{1}}{\mathrm{2}}\:\:,\:{squaring} \\ $$$$\mathrm{1}+\mathrm{sin}\:\mathrm{2}\theta=\frac{\mathrm{1}}{\mathrm{4}}\:\:\Rightarrow\:\:\mathrm{sin}\:\mathrm{2}\theta=−\frac{\mathrm{3}}{\mathrm{4}}\:\:. \\ $$$$ \\ $$$$ \\ $$

Commented by Tinkutara last updated on 20/May/17

In (i) shouldn′t it be something else because cos ((π/2) + θ) = − sin θ?

$$\mathrm{In}\:\left(\mathrm{i}\right)\:\mathrm{shouldn}'\mathrm{t}\:\mathrm{it}\:\mathrm{be}\:\mathrm{something}\:\mathrm{else} \\ $$$$\mathrm{because}\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}\:+\:\theta\right)\:=\:−\:\mathrm{sin}\:\theta? \\ $$

Commented by ajfour last updated on 20/May/17

i have obtained +(3/4) as well.

$${i}\:{have}\:{obtained}\:+\frac{\mathrm{3}}{\mathrm{4}}\:{as}\:{well}. \\ $$

Commented by ajfour last updated on 20/May/17

if sin A=cos B we conclude A=(π/2)+B , here in (i)

$${if}\:\mathrm{sin}\:{A}=\mathrm{cos}\:{B} \\ $$$${we}\:{conclude}\:\:{A}=\frac{\pi}{\mathrm{2}}+{B}\:,\:{here}\:{in}\:\left({i}\right) \\ $$

Commented by Tinkutara last updated on 20/May/17

Thanks!

$$\mathrm{Thanks}! \\ $$

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