Question

What is the distance between the points \[A\left(\sin\theta - \cos\theta, 0 \right)\] and \[B\left(0, \sin\theta + \cos\theta \right)\] ?

Answer 1

The given points are \[A\left( \sin\theta - \cos\theta, 0 \right)\] and \[B\left( 0, \sin\theta + \cos\theta \right)\].

Using distance formula, we have

\[AB = \sqrt{\left[ \left( \sin\theta - \cos\theta \right) - 0 \right]^2 + \left[ 0 - \left( \sin\theta + \cos\theta \right) \right]^2}\]
\[ = \sqrt{\left( \sin\theta - \cos\theta \right)^2 + \left( \sin\theta + \cos\theta \right)^2}\]
\[ = \sqrt{\sin^2 \theta + \cos^2 \theta - 2\sin\theta\cos\theta + \sin^2 \theta + \cos^2 \theta + 2\sin\theta\cos\theta}\]
\[ = \sqrt{2\left( \sin^2 \theta + \cos^2 \theta \right)}\]
\[ = \sqrt{2} \text{ units }  \left( \sin^2 \theta + \cos^2 \theta = 1 \right)\]

Thus, the distance between the given points is \[\sqrt{2}\] units.