Question

find the exact value of \\( \sin m \\) in simplest radical form.

Explanation:

Step1: Recall sine definition for right triangles

For an acute angle in a right triangle, $\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$

Step2: Identify sides for $\angle M$

• Opposite side to $\angle M$: $ON = 2$
• Hypotenuse: $OM = \sqrt{18} = 3\sqrt{2}$

Step3: Substitute into sine formula

$\sin M = \frac{2}{\sqrt{18}}$

Step4: Simplify the radical fraction

First simplify $\sqrt{18} = 3\sqrt{2}$, then rationalize:
$\sin M = \frac{2}{3\sqrt{2}} = \frac{2\sqrt{2}}{3\sqrt{2} \cdot \sqrt{2}} = \frac{2\sqrt{2}}{6} = \frac{\sqrt{2}}{3}$

Answer:

$\frac{\sqrt{2}}{3}$