Question

when the axes are rotated through an angle 30 find the new coordinates of the following point (-2,4)
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Answer 1

x' = x cos(θ) + y sin(θ)

y' = - x sin(θ) + y cos(θ)

So,

x' = -2(√3/2) + 4(1/2) = 2 - √3

y' = 2(1/2) + (4)(√3/2) = 1 + 2√3

New coordinates = (2 - √3, 1 + 2√3)

Answer 2

Answer:

The new coordinates are  (x',y') = (2-√3, 1+2√3)

Step-by-step explanation:

Given coordinates  (-2,4)

The angle of rotation θ = 30°  

Here we need to find new coordinates of the given point (-2,4)  

Let the coordinates (x,y) are rotated through an angle θ then the coordinates of new points are (x', y')

where x' = x cos θ + y sin θ

           y' = - x sin θ + y cos θ    

Let given points (-2,4) = (x,y); rotated through an angle θ = 30°

Therefore, new coordinates

 ⇒ x' = (-2) cos 30° + 4 sin 30°            

         = $-2(\frac{\sqrt{3} }{2} )+ 4 (\frac{1}{2})$          [ ∵ cos 30° = $\frac{\sqrt{3} }{2}$ ,  sin 30° = $\frac{1}{2}$ ]

         = - √3 + 2  

         =  2-√3      

⇒  y' = -(-2) sin 30° + 4 cos 30°  

        =  $2(\frac{1}{2} ) + 4(\frac{\sqrt{3} }{2} )$  

        =  1 + 2√3  

The new coordinates are  (x',y') = (2-√3, 1+2√3)

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