The coordinate axes are rotated throughan angle ${135}^0$. If the  coordinates of a point P in  the new system are known to be (4, –3) then the coordinates P in the original system are

$$\begin{gathered} \mathrm{Given} \mathrm{\theta }={135}^0 \mathrm{and} (\mathrm{x},\mathrm{y})=(4,-3) \\ \mathrm{let} (\mathrm{x},\mathrm{y}) \mathrm{be} \mathrm{original} \mathrm{coordinates} \\ \mathrm{then} \mathrm{x}=X \cos \mathrm{\theta }-Y \sin \mathrm{\theta } \\ =4 \cos {135}^0-(-3) \sin {135}^0 \\ =4\left(\frac{-1}{\sqrt{2}}\right)+3\left(\frac{1}{\sqrt{2}}\right) \\ =\frac{-1}{\sqrt{2}} \\ \mathrm{and} \mathrm{y}=X \sin \mathrm{\theta }+Y \cos \mathrm{\theta } \\ =4 \sin {135}^0-3 \cos {135}^0 \\ =4\left(\frac{1}{\sqrt{2}}\right)-3\left(\frac{-1}{\sqrt{2}}\right) \\ =\frac{4}{\sqrt{2}}+\frac{3}{\sqrt{2}} \\ =\frac{7}{\sqrt{2}} \\ \therefore (\mathrm{x},\mathrm{y}) = \left(\frac{-1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right) \end{gathered}$$