Given a point ( x, y ), what is the new point when the point is rotated counterclockwise about the origin 270 degrees

Answer 1

To rotate a point $(x,y)$ counterclockwise about the origin by $270$ degrees, you can use the rotation transformation formulas.

The general formula for rotating a point $(x,y)$ counterclockwise by an angle $\theta$ is given by:

$$(x^{\prime },y^{\prime })=(x\cos \theta -y\sin \theta ,x\sin \theta +y\cos \theta )$$

For a $270$ degrees rotation, we first convert this angle to radians:

$$\theta ={270}^{\circ }=\frac{3\pi }{2}\text{ radians}$$

Now, we find the cosine and sine of $270$ degrees:

$$\cos ({270}^{\circ })=0$$ $$\sin ({270}^{\circ })=-1$$

Substituting these values into the rotation formulas:

$$x^{\prime }=x\cdot 0-y\cdot (-1)=y$$ $$y^{\prime }=x\cdot (-1)+y\cdot 0=-x$$

Thus, the new coordinates after rotating $(x,y)$ by $270$ degrees counterclockwise are:

$$(x^{\prime },y^{\prime })=(y,-x)$$

So the new point after the rotation is $(y,-x)$.