Question

When the axes are rotated to an angle of 60° the point P is changed as (3,4).
Find P.​ pls help me to this question I will follow and mark as brain list

Answer 1

Answer:

Coordinates are : [(3 - 4√3)/2,  (4 + 3√3)/2]

Step-by-step explanation:

If the coordinate axes rotates at an angle θ. Then original coordinate P(x,y)  change into P'(x',y')

x = x'cosθ - y'sinθ

y = x'sinθ + y'cosθ

Given that a point changes to (3,4).

Here x' = 3 and y'= 4 and θ = 60°

Let (x,y) be the original coordinate of P

Then

x = 3 cos 60° - (4) sin 60°

  = 3(1/2) - 4(√3/2)

 = (3/2) - 2√3

 = (3 - 4√3)/2

y = 3 sin 60° + 4 cos 60°

  = 3(√3/2) + 4(1/2)

  = (4 + 3√3)/2

Therefore,

Coordinates are : [(3 - 4√3)/2,  (4 + 3√3)/2]

Hope it helps!

Answer 2

Answer:

P {(3- 4√3)/2 , (3√3 +4)/2}

Step-by-step explanation:

Concept= Rotation of axes

Given= The rotation angle and the new coordinates

To find= The original Coordinates before rotation

Explanation=

We have been the question as when the axes are rotated to an angle of 60° the point P is changed as (3,4).We are told to find the Coordinates of P.

Rotation of axes is done by following a procedure which is formula.

The new coordinates are taken as (X,Y) and the old coordinates are taken as (x,y) on which rotation is carried out.

Here the angle of rotation is taken as θ

Formula=.

                    X        Y

          x       cosθ   -sinθ

          y       sinθ     cosθ

x= Xcosθ - Ysinθ

y= Xsinθ  + Ycosθ

So we know that new coordinates P are(X,Y) which is (3,4)

and the initial P coordinates are (x,y) . Here the angle of rotation θ is 60°

So,

x= 3cos60° - 4sin60°

y=3sin60° + 4cos60°

x= 3*1/2 -4*√3/2 = (3- 4√3)/2

y= 3*√3/2 + 4*1/2 = (3√3 + 4)/2

So the coordinates of P are (3- 4√3)/2 , (3√3 +4)/2

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