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Question Number 35362 by 26488679 last updated on 18/May/18

Commented by 26488679 last updated on 18/May/18

is arcsin(−(1/2)) can be  equal to 210°?

$${is}\:{arcsin}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\:{can}\:{be}\:\:{equal}\:{to}\:\mathrm{210}°? \\ $$

Commented by ajfour last updated on 18/May/18

not a principal value.

$${not}\:{a}\:{principal}\:{value}. \\ $$

Commented by prof Abdo imad last updated on 18/May/18

we have arcsin(−(1/2))=−(π/6) and arcsin(((√3)/2))=(π/3)  ⇒sin{arcsin(−(1/2)) +2 arcsin(((√3)/2))}=  sin(−(π/6) +((2π)/3)) =sin( ((3π)/6))=sin((π/2))=1  answerD

$${we}\:{have}\:{arcsin}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)=−\frac{\pi}{\mathrm{6}}\:{and}\:{arcsin}\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)=\frac{\pi}{\mathrm{3}} \\ $$$$\Rightarrow{sin}\left\{{arcsin}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\:+\mathrm{2}\:{arcsin}\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)\right\}= \\ $$$${sin}\left(−\frac{\pi}{\mathrm{6}}\:+\frac{\mathrm{2}\pi}{\mathrm{3}}\right)\:={sin}\left(\:\frac{\mathrm{3}\pi}{\mathrm{6}}\right)={sin}\left(\frac{\pi}{\mathrm{2}}\right)=\mathrm{1}\:\:{answerD} \\ $$

Answered by ajfour last updated on 18/May/18

sin (−(π/6)+2×(π/3))=sin (π/2) .

$$\mathrm{sin}\:\left(−\frac{\pi}{\mathrm{6}}+\mathrm{2}×\frac{\pi}{\mathrm{3}}\right)=\mathrm{sin}\:\frac{\pi}{\mathrm{2}}\:. \\ $$

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