Answer
a) $T(4,4)=(0,4\sqrt{2}) $
b) $T(4,4)=[2\sqrt{3}-2,2+2\sqrt{3}]$
c) $T(5,0)=((\frac{-5}{2}) ,{ }5(\frac{\sqrt{3}}{2}) )$
Work Step by Step
$T(x,y)=(xcos\theta -ysin\theta ,{ }xsin\theta +ycos\theta) $
a) When $\theta =45^{\circ}$
$T(4,4)=(4cos( 45^{\circ})-4sin(45^{\circ}) ,{ }4sin(45^{\circ}) +4cos(45^{\circ})) \\
=[4(\frac{1}{\sqrt{2}})-4(\frac{1}{\sqrt{2}}) ,{ }4(\frac{1}{\sqrt{2}}) +4(\frac{1}{\sqrt{2}})] \\
=[2\sqrt{2}-2\sqrt{2},2\sqrt{2}+2\sqrt{2}]\\
=(0,4\sqrt{2}) $
b) When $\theta =30^{\circ}$
$T(4,4)=(4cos( 30^{\circ})-4sin(30^{\circ}) ,{ }4sin(30^{\circ}) +4cos(30^{\circ})) \\
=[4(\frac{\sqrt{3}}{2})-4(\frac{1}{2}) ,{ }4(\frac{1}{{2}}) +4(\frac{\sqrt{3}}{2})] \\
=[2\sqrt{3}-2,2+2\sqrt{3}]$
c) When $\theta =120^{\circ}$
$T(5,0)=(5cos( 120^{\circ})-0sin(120^{\circ}) ,{ }5sin(120^{\circ}) +0cos(120^{\circ})) \\
=(5(\frac{-1}{2}) ,{ }5(\frac{\sqrt{3}}{2}) )\\
=((\frac{-5}{2}) ,{ }5(\frac{\sqrt{3}}{2}) )$
