In a right triangle, the length of the side opposite an angle can be determined using the sine function. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
For the first triangle, we have:
\[ \sin(35^\circ) = \frac{4}{18} \]
Now, we can calculate the sine of \(35^\circ\):
First, calculate:
\[ \frac{4}{18} = \frac{2}{9} \]
Thus,
\[ \sin(35^\circ) \approx 0.5736 \quad (\text{using a calculator}) \]
This means that the sine of \(35^\circ\) matches our earlier calculation. Now let's use this value for the second triangle which has a hypotenuse of \(9\) and we want to find the opposite side:
Using the sine formula again:
\[ \sin(35^\circ) = \frac{\text{opposite}}{9} \]
Now plug in the value of \(\sin(35^\circ)\):
\[ 0.5736 = \frac{\text{opposite}}{9} \]
To find the length of the side opposite the \(35^\circ\) angle, we multiply both sides by \(9\):
\[ \text{opposite} = 9 \times 0.5736 \approx 5.1624 \]
Thus, the length of the side opposite the \(35^\circ\) angle in the second triangle is approximately:
\[ \boxed{5.16} \text{ units} \]
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