Answer:
[tex]\dfrac{\sqrt{2}}{2}[/tex]
Step-by-step explanation:
The trigonometric ratio sine is defined as:
[tex]\sin(\theta)=\dfrac{\text{opposite}}{\text{hypotenuse}}[/tex]
From the diagram, we can identify the following side lengths in relation to [tex]\theta[/tex]:
- [tex]\text{opposite} = \sqrt3[/tex]
- [tex]\text{adjacent} = \sqrt6[/tex]
Plugging these into the sine ratio:
[tex]\sin(\theta)=\dfrac{\sqrt3}{\sqrt6}[/tex]
We can rewrite this by rationalizing the denominator:
[tex]\dfrac{\sqrt3}{\sqrt6} \cdot \dfrac{\sqrt6}{\sqrt6} = \dfrac{\sqrt{18}}6[/tex]
Then, the numerator can be simplified by prime factorizing the number under the square root and taking out a perfect square of 3:
[tex]\dfrac{\sqrt{3\cdot 3 \cdot 2}}{6}[/tex]
[tex]= \dfrac{3\sqrt2}{6}[/tex]
[tex]\boxed{=\dfrac{\sqrt{2}}{2}}[/tex]
