The coordinates of the point in the&nbsp;x'&nbsp;y'&nbsp;- coordinate system with the given angle of rotation and the xy-coordinates.

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The coordinates of the point in the&amp;nbsp;x&#039;&amp;nbsp;y&#039;&amp;nbsp;- coordinate system with the given angle of rotation and the xy-coordinates.

lamanocornudaW · Accepted Answer

Suppose the x- and y- axes are rotated about the origin through a positive acute angle θ, then the coordinates (x, y) and (x′,y′) of a point P in the xy- and x′,y′- coordinate systems are related by the following formulas:x′=xcos⁡θ + ysin⁡θy′= −xsin⁡θ + ycos⁡θx=x′cos⁡θ − y′sin⁡θy=x′sin⁡θ + y′cos⁡θGiven:The angle of rotation is θ=30∘ and the x- and y- coordinates are 0 and 2, respectively.Calculation:Use the definition, substitute the values of x-, y- coordinates and θ in order to obtain the values of x′, y′ - coordinates.x′=(0)cos⁡ 30∘ + (2)sin⁡ 30∘y′= −(0)sin⁡ 30∘ + (2)cos⁡ 30∘Know that sin⁡ 30∘=12 and 30∘=32Thus, the x′ and y′ - coordinates become x′=(0) (32) + (2)(12)=0 + (1)=1y′= −(0)(12) + (2)(32)=0 + (3)=3Therefore, the coordinates of the point the x′y′ - coordinate system are 1 and 3, respectively.

The coordinates of the point in the x^{prime} y^{prime} - coordinate system with the given angle of rotation and the xy-coordinates.

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