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 θ.

$$θ=\frac{π}{3}$$

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Bentley Leach

Use the axis rotation formulas: $$x=u\cosθ-v\sinθ$$

$$y=u\sinθ+v\cosθ$$

Substitute $$θ=\frac{π}{3}$$. This is a special angle so we can find the exact values.

For x, $$x=u\cos \frac{π}{3}-v\sin \frac{π}{3}$$

$$x=\frac{u}{2}-\sqrt{\frac{3v}{2}}$$

For y, $$\displaystyle{y}={u} \sin{{\left(\frac{\pi}{{3}}\right)}}+{v} \cos{{\left(\frac{\pi}{{3}}\right)}}$$

$$\displaystyle{y}=\sqrt{{\frac{{{3}{u}}}{{2}}}}+\frac{v}{{2}}$$