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Question Number 140666 by john_santu last updated on 11/May/21

Find the coordinates of (−2,0) when the axes are rotated counterclockwise through the angle arcsin (4/5).

$${Find}\:{the}\:{coordinates}\:{of}\:\left(−\mathrm{2},\mathrm{0}\right)\: \\ $$$${when}\:{the}\:{axes}\:{are}\:{rotated}\:{counterclockwise} \\ $$$${through}\:{the}\:{angle}\:\mathrm{arcsin}\:\frac{\mathrm{4}}{\mathrm{5}}. \\ $$

Answered by bemath last updated on 11/May/21

{ ((x′=x cos θ−ysin θ)),((y′=xsin θ+ycos θ)) :} { ((x′=−2((3/5)))),((y=−2((4/5)))) :} →(x′,y′) = (−(6/5),−(8/5))

$$\begin{cases}{\mathrm{x}'=\mathrm{x}\:\mathrm{cos}\:\theta−\mathrm{ysin}\:\theta}\\{\mathrm{y}'=\mathrm{xsin}\:\theta+\mathrm{ycos}\:\theta}\end{cases} \\ $$$$\begin{cases}{\mathrm{x}'=−\mathrm{2}\left(\frac{\mathrm{3}}{\mathrm{5}}\right)}\\{\mathrm{y}=−\mathrm{2}\left(\frac{\mathrm{4}}{\mathrm{5}}\right)}\end{cases}\:\rightarrow\left(\mathrm{x}',\mathrm{y}'\right)\:=\:\left(−\frac{\mathrm{6}}{\mathrm{5}},−\frac{\mathrm{8}}{\mathrm{5}}\right) \\ $$

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