Find the linear equations that can be used to convert

geduiwelh 2021-08-14 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 \(\theta\). \(\theta=\tan^{-1}(\frac{5}{12})\)

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Expert Answer

l1koV
Answered 2021-08-15 Author has 2895 answers

Use the axis rotation formulas:

\(x=u \cos \theta-v \sin \theta\)

\(y=u \sin \theta+v \cos \theta\)

From the given angle, we know that: \(\tan \theta=\frac{5}{12}\)

From this tangent ration, opp=5 and adj=12 so \(hyp=\sqrt{((opp)^2+(adj)^2)}=\sqrt{(5^2+12^2)}=\sqrt{169}=13\) so: \(\sin \theta=\frac{opp}{hyp}=\frac{5}{12}\)

and

\(\cos \theta=\frac{adj}{hyp}=\frac{12}{13}\)

 For x, \(x=\frac{12u}{13}-\frac{5v}{13}\) For y, \(y=\frac{5u}{13}+\frac{12v}{13}\)

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