Find the linear equations that can be used to convert

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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l1koV

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, $$y=\frac{5u}{13}+\frac{12v}{13}$$