Question

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

Alternate coordinate systems
ANSWERED
asked 2020-11-17

The coordinates of the point in the \(\displaystyle{x}' {y}'\) - coordinate system with the given angle of rotation and the xy-coordinates.

Answers (1)

2020-11-18

Suppose the x- and y- axes are rotated about the origin through a positive acute angle \(\displaystyle\theta\), then the coordinates \(\displaystyle{\left({x},\ {y}\right)}\) and \(\displaystyle{\left({x}', {y}'\right)}\) of a point P in the xy- and \(\displaystyle{x}',{y}'\)- coordinate systems are related by the following formulas:
\(\displaystyle{x}'={x}{\cos{\theta}}\ +\ {y}{\sin{\theta}}\)
\(\displaystyle{y}'=\ -{x}{\sin{\theta}}\ +\ {y}{\cos{\theta}}\)
\(\displaystyle{x}={x}'{\cos{\theta}}\ -\ {y}'{\sin{\theta}}\)
\(\displaystyle{y}={x}'{\sin{\theta}}\ +\ {y}'{\cos{\theta}}\)
Given:
The angle of rotation is \(\displaystyle\theta={30}^{{\circ}}\) and the x- and y- coordinates are 0 and 2, respectively.
Calculation:
Use the definition, substitute the values of x-, y- coordinates and \(\displaystyle\theta\) in order to obtain the values of \(\displaystyle{x}',\ {y}'\) - coordinates.
\(\displaystyle{x}'={\left({0}\right)}{\cos{\ }}{30}^{{\circ}}\ +\ {\left({2}\right)}{\sin{\ }}{30}^{{\circ}}\)
\(\displaystyle{y}'=\ -{\left({0}\right)}{\sin{\ }}{30}^{{\circ}}\ +\ {\left({2}\right)}{\cos{\ }}{30}^{{\circ}}\)
Know that \(\displaystyle{\sin{\ }}{30}^{{\circ}}={\frac{{{1}}}{{{2}}}}\ {\quad\text{and}\quad}\ {30}^{{\circ}}={\frac{{\sqrt{{{3}}}}}{{{2}}}}\)
Thus, the \(\displaystyle{x}'\ {\quad\text{and}\quad}\ {y}'\) - coordinates become \(\displaystyle{x}'={\left({0}\right)}\ {\left({\frac{{\sqrt{{{3}}}}}{{{2}}}}\right)}\ +\ {\left({2}\right)}{\left({\frac{{{1}}}{{{2}}}}\right)}\)
\(\displaystyle={0}\ +\ {\left({1}\right)}\)
=1
\(\displaystyle{y}'=\ -{\left({0}\right)}{\left({\frac{{{1}}}{{{2}}}}\right)}\ +\ {\left({2}\right)}{\left({\frac{{\sqrt{{{3}}}}}{{{2}}}}\right)}\)
\(\displaystyle={0}\ +\ {\left(\sqrt{{{3}}}\right)}\)
\(\displaystyle=\sqrt{{{3}}}\)
Therefore, the coordinates of the point the \(\displaystyle{x}' {y}'\) - coordinate system are 1 and \(\displaystyle\sqrt{{{3}}}\), respectively.

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