# Find the linear equations that can be used to convert an (x, y) equation to a (x, v) equation using the given angle of rotation \displaystyle\theta. \displaystyle\theta={{\tan}^{{-{1}}}{\left({5}\text{/}{12}\right)}}

Annette Arroyo 2021-01-04 Answered

Find the linear equations that can be used to convert an (x, y) equation to a (x, v) equation using the given angle of rotation $\theta$.
$\theta =\left\{{\mathrm{tan}}^{-1}\frac{5}{12}\right\}$

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lamusesamuset

Use the axis rotation formulas:
$x=u\mathrm{cos}\theta -v\mathrm{sin}\theta$
$y=u\mathrm{sin}\theta +v\mathrm{cos}\theta$
From the given angle, we know that:
$\mathrm{tan}\theta =\frac{5}{12}$
From this tangent ratio, $opp=5$ and $adj=12$ so

$hyp=\sqrt{\left(opp{\right)}^{2}+\left(adj{\right)}^{2}}=\sqrt{{5}^{2}+{12}^{2}}=\sqrt{169}=13$ so:
$\mathrm{sin}\theta =\frac{opp}{hyp}=\frac{5}{12}$
and
$\mathrm{cos}\theta =\frac{adj}{hyp}=\frac{12}{13}$
For x,
$x=\frac{12u}{13}-\frac{5v}{13}$
For y,
$y=\frac{5u}{13}+\frac{12v}{13}$