The point (8,0) is rotated about the origin counterclockwise by 38 degrees. Determine the coordinates of the image.

Question

We got a bunch of questions that were considered hard for the section of Geometry that we are on as of right now, and I answered most of them with ease. But, this one I could not answer. Any help would be appreciated.

The point (8,0) is rotated about the origin counterclockwise by 38 degrees. Determine the coordinates of the image.

Answer 1

Draw a picture. Call the point that $(8,0)$ is rotated to by the name $P=(a,b)$. We want to find $a$ and $b$.

Call the origin $O$. Call the point on the $x$-axis that is directly below $P$ by the name $Q$. Note that $a$ and $b$ are positive. Note also that $a=OQ$ and $b=QP$.

Look at the right triangle $OPQ$. The hypotenuse $OP$ is equal to $8$. You know $\angle POQ$. Now you can use "trigonometry" to find $OQ$ and $PQ$.

Answer 2

• The new point is at a distance $r = 8$ away from the origin.
• The new point is at an angle $\theta = 38^\circ$ counterclockwise from the positive $x$-axis.

In general, if you know these two values ($r$ and $\theta$) of a point, the coordinates of that point are $x = r \cos \theta$, and $y = r \sin \theta$. You can see this by drawing a triangle with hypotenuse going from $(0,0)$ to the given point.

Answer 3

For any angle $\theta$ the matrix,

$$ \left( \begin{array}{cc}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \\\end{array} \right)$$

gives a rotation by $\theta$ via vector multiplication.