Question

When the axes rotated through angle of 60point p change as 3,4 find the coordinates

Answer 1

        $\text{\huge \bf \underline{Answer:}}$

        $\text{It is given that when the axes is rotated through an angle of }60^{\circ}\\\text{point P changes to (3,4)}$

        $\text{Let the original points be }(x,y) \text{ and the new points be }(x',y') = (3,4)$

        $\text{We know that }x = x'cos\theta - y'sin\theta \text{ and }y = x'sin\theta + y'cos\theta$

$\implies \: \theta = 60^{\circ}$

$\implies \: x = (3\times cos60) - (4 \times sin60) \; \; ; \; \; y = (3 \times sin60) + (4\times cos60)$

$\implies \: x = \text{\huge{(}}3\times \dfrac12\text{\huge{)}} - \text{\huge{(}}4 \times \dfrac{\sqrt3}{2}\text{\huge{)}} \; \; ; \; \; y = \text{\huge{(}}3 \times \dfrac{\sqrt3}{2}\text{\huge{)}} + \text{\huge{(}}4\times \dfrac12\text{\huge{)}}$

$\implies \: x = \dfrac32 - \dfrac{4\sqrt3}{2} \; \; ; \; \; y = \dfrac{3\sqrt3}{2} + \dfrac42$

$\implies \: \boxed{\boxed{x = \dfrac{3-4\sqrt3}{2} \; \; ; \; \; y = \dfrac{4+3\sqrt3}{2}}}$