First slide

Cartesian plane

Question

If the axes are rotated through an angle of $30 °$ in clockwise direction, the point $(4, 2 \sqrt{3})$ in the new system is

Moderate

A

$(2, 3)$
B

$(2, \sqrt{3})$
C

$(\sqrt{3}, 2)$
D

$(\sqrt{3}, 5)$

Solution

We have $x = 4, y = 2 \sqrt{3}$ and $θ = - 30^{\circ}$
$∴ X = x \cos θ + y \sin θ$ and $Y = - x \sin θ + y \cos θ$
$\Rightarrow X = 4 \cos 30^{\circ} - 2 \sqrt{3} \sin 30^{\circ}$
$Y = + 4 \sin 30^{\circ} + 2 \sqrt{3} \cos 30^{\circ}$
$= X = \sqrt{3} and Y = 5$
Hence, the given point in the new system is $(\sqrt{3}, 5)$

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