To find the length of side [tex]\( b \)[/tex] in a right triangle, given the opposite side and the angle, we can use the trigonometric relationship involving the tangent function:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Here, we are given:
- [tex]\(\theta = 55^\circ\)[/tex]
- The length of the opposite side = 15 cm
We need to find the length of the adjacent side [tex]\( b \)[/tex]. Rearranging the tangent formula to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \frac{\text{opposite}}{\tan(\theta)}
\][/tex]
Substituting the given values:
[tex]\[
b = \frac{15}{\tan(55^\circ)}
\][/tex]
After calculating the above expression, we get [tex]\( b \approx 10.503 \text{ cm} \)[/tex].
Now, let's compare this result to the given options to find the correct answer:
- [tex]\(3.0 \text{ cm}\)[/tex]
- [tex]\(9.8 \text{ cm}\)[/tex]
- [tex]\(10.5 \text{ cm}\)[/tex]
- [tex]\(12.8 \text{ cm}\)[/tex]
The closest option to [tex]\(10.503 \text{ cm}\)[/tex] is [tex]\(10.5 \text{ cm}\)[/tex].
Therefore, the length of side [tex]\( b \)[/tex] is approximately [tex]\( \boxed{10.5 \text{ cm}} \)[/tex].
