When the coordinate axes are rotated through an angle 135 ° , the coordinates of a point P in the new…

Let, we have a point $P\left(x, y\right)$, 

Now, the coordinate axes rotated through an angle $135°$, and the coordinate of the point $P$ in new system is $\left(4, -3\right)$.

Hence, applying the formula, the coordinates of the point $P$ in the original system is 

$x=x^{\text{'}}\cos \theta -y^{\text{'}}\sin \theta =4\cos 135°-\left(-3\right)\sin 135°=-\frac{4}{\sqrt{2}}+\frac{3}{\sqrt{2}}=-\frac{1}{\sqrt{2}}$Similarly, 

$y=x^{\text{'}}\sin \theta +y^{\text{'}}\cos \theta =4\sin 135°+\left(-3\right)\cos 135°=\frac{4}{\sqrt{2}}+\frac{3}{\sqrt{2}}=\frac{7}{\sqrt{2}}$

where, $\left(x, y\right)$ is the coordinate of original system and $\left(x^{\text{'}} ,y^{\text{'}}\right)$ is the coordinate of new system.