Question

Find the distance between the given pairs of point $(\cos \theta ,-\sin \theta )$ and $(\sin \theta ,-\cos \theta )$

Answer 1

We know that the distance between the two points $(x_1,y_1)$ and $(x_2,y_2)$ is

$d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Here, the given points are $(\cos \theta -\sin \theta )$ and $(\sin \theta -\cos \theta )$, therefore,

$d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}=\sqrt{{(\sin \theta -\cos \theta )}^2+{(-\cos \theta -(-\sin \theta \left)\right)}^2}$
$=\sqrt{{(\sin \theta -\cos \theta )}^2+{(\sin \theta -\cos \theta )}^2}$

$=\sqrt{{2(\sin \theta -\cos \theta )}^2}=\sqrt{2}(\sin \theta -\cos \theta )$

Hence, the distance is $\sqrt{2}(\sin \theta -\cos \theta )$ units.