Question

find the sine of $\angle v$.

write your answer in simplified, rationalized form. do not round.
$\sin(v) = $

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 $\angle V$

Opposite side to $\angle V$: $XW = 3\sqrt{3}$
First, calculate hypotenuse $VW$ using Pythagorean theorem:
$$VW = \sqrt{(3\sqrt{2})^2 + (3\sqrt{3})^2} = \sqrt{18 + 27} = \sqrt{45} = 3\sqrt{5}$$

Step3: Compute $\sin(V)$

Substitute values into the sine formula:
$$\sin(V) = \frac{3\sqrt{3}}{3\sqrt{5}} = \frac{\sqrt{3}}{\sqrt{5}}$$
Rationalize the denominator:
$$\frac{\sqrt{3}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{15}}{5}$$

Answer:

$\frac{\sqrt{15}}{5}$