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.
The coordinates of the point in the x^{prime} y^{prime} - coordinate system with the given angle of rotation and the xy-coordinates.
Answers (1)
Relevant Questions
Solve the given Alternate Coordinate Systems and give a correct answer 10) Convert the equation from Cartesian to polar coordinates solving for \(r^2\):
\(\frac{x^2}{9} - \frac{y^2}{16} = 25\)
Given the elow bases for \(R^2\) and the point at the specified coordinate in the standard basis as below, (40 points)
\((B1 = \left\{ (1, 0), (0, 1) \right\} \)&
\( B2 = (1, 2), (2, -1) \}\)(1, 7) = \(3^* (1, 2) - (2, 1)\)
\(B2 = (1, 1), (-1, 1) (3, 7 = 5^* (1, 1) + 2^* (-1,1)\)
\(B2 = (1, 2), (2, 1) \ \ \ (0, 3) = 2^* (1, 2) -2^* (2, 1)\)
\((8,10) = 4^* (1, 2) + 2^* (2, 1)\)
B2 = (1, 2), (-2, 1) (0, 5) =
(1, 7) =
a. Use graph technique to find the coordinate in the second basis. (10 points) b. Show that each basis is orthogonal. (5 points) c. Determine if each basis is normal. (5 points) d. Find the transition matrix from the standard basis to the alternate basis. (15 points)
To determine:
a) Whether the statement, " The point with Cartesian coordinates \(\displaystyle{\left[\begin{array}{cc} -{2}&\ {2}\end{array}\right]}\) has polar coordinates \(\displaystyle{\left[{b}{f}{\left({2}\sqrt{{{2}}},\ {\frac{{{3}\pi}}{{{4}}}}\right)}\ {\left({2}\sqrt{{{2}}},{\frac{{{11}\pi}}{{{4}}}}\right)}\ {\left({2}\sqrt{{{2}}},\ -{\frac{{{5}\pi}}{{{4}}}}\right)}\ {\quad\text{and}\quad}\ {\left(-{2}\sqrt{{2}},\ -{\frac{{\pi}}{{{4}}}}\right)}\right]}\) " is true or false.
b) Whether the statement, " the graphs of \(\displaystyle{\left[{r}{\cos{\theta}}={4}\ {\quad\text{and}\quad}\ {r}{\sin{\theta}}=\ -{2}\right]}\) intersect exactly once " is true or false.
c) Whether the statement, " the graphs of \(\displaystyle{\left[{r}={4}\ {\quad\text{and}\quad}\ \theta={\frac{{\pi}}{{{4}}}}\right]}\) intersect exactly once ", is true or false.
d) Whether the statement, " the point \(\displaystyle{\left[\begin{array}{cc} {3}&{\frac{{\pi}}{{{2}}}}\end{array}\right]}{l}{i}{e}{s}\ {o}{n}\ {t}{h}{e}\ {g}{r}{a}{p}{h}\ {o}{f}{\left[{r}={3}{\cos{\ }}{2}\ \theta\right]}\) " is true or false.
e) Whether the statement, " the graphs of \(\displaystyle{\left[{r}={2}{\sec{\theta}}\ {\quad\text{and}\quad}\ {r}={3}{\csc{\theta}}\right]}\) are lines " is true or false.
To find: The equivalent polar equation for the given rectangular-coordinate equation.
Given:
\(\displaystyle{x}^{2}+{y}^{2}+{8}{x}={0}\)
