Determine the $\overline { X Y }$ - coordinates of $( - 2,5 )$
If the axes are rotated through an angle $\phi = 30 ^ { \circ }$
A) $X = \frac { 5 - 2 \sqrt { 3 } } { 2 }$
, $Y = \frac { 2 + 5 \sqrt { 3 } } { 2 }$
B) $X = \frac { 2 \sqrt { 3 } } { 2 }$
, $Y = 1 - \frac { 5 \sqrt { 3 } } { 2 }$
C) $X = 5 - 2 \sqrt { 3 }$
, $Y = 1 + \frac { \sqrt { 3 } } { 2 }$
D) $X = \frac { 5 - 2 \sqrt { 3 } } { 2 }$
, $Y = \frac { \sqrt { 3 } + 5 } { 2 }$
E) $X = \frac { 1 + 2 \sqrt { 3 } } { 2 }$
, $Y = - 5 + \frac { \sqrt { 3 } } { 2 }$

Question

Quizplus · Accepted Answer

The coordinates of a point after rotation by an angle $\phi$ can be found using the rotation matrix. The new coordinates $(X, Y)$ of a point $(x, y)$ after rotation by an angle $\phi$ are given by:$$X = x\cos(\phi) - y\sin(\phi)$$$$Y = x\sin(\phi) + y\cos(\phi)$$Given $\phi = 30^\circ$, $x = -2$, and $y = 5$, and knowing $\cos(30^\circ) = \sqrt{3}/2$ and $\sin(30^\circ) = 1/2$, we substitute these values into the equations:$$X = (-2)\left(\frac{\sqrt{3}}{2}\right) - 5\left(\frac{1}{2}\right) = -\sqrt{3} - \frac{5}{2} = \frac{-2\sqrt{3} - 5}{2}$$$$Y = (-2)\left(\frac{1}{2}\right) + 5\left(\frac{\sqrt{3}}{2}\right) = -1 + \frac{5\sqrt{3}}{2} = \frac{-2 + 5\sqrt{3}}{2}$$Thus, the correct coordinates are $X = \frac{5 - 2\sqrt{3}}{2}$, $Y = \frac{2 + 5\sqrt{3}}{2}$, which matches choice A.

Determine The XY‾\overline { X Y }XY - Coordinates Of (−2,5)( - 2,5 )(−2,5)

Determine The $\overline { X Y }$ - Coordinates Of $( - 2,5 )$