In triangle \(ABC\), we have:
- Angle \(A = 14^\circ\)
- Angle \(B = 52^\circ\)
- Side \(c = 17\) cm (opposite angle \(C\))
First, we need to find angle \(C\):
\[ C = 180^\circ - A - B = 180^\circ - 14^\circ - 52^\circ = 114^\circ \]
Now we can use the Law of Sines, which states:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
We want to find the length of side \(b\):
\[ \frac{b}{\sin B} = \frac{c}{\sin C} \]
Substituting the known values:
\[ \frac{b}{\sin 52^\circ} = \frac{17}{\sin 114^\circ} \]
Now, we can calculate \(\sin 52^\circ\) and \(\sin 114^\circ\):
- \(\sin 52^\circ \approx 0.7880\)
- \(\sin 114^\circ \approx \sin(180^\circ - 66^\circ) = \sin 66^\circ \approx 0.9135\)
Now substitute these values into the equation:
\[ \frac{b}{0.7880} = \frac{17}{0.9135} \]
Now, solve for \(b\):
\[ b = 17 \times \frac{0.7880}{0.9135} \]
Calculating the right-hand side:
\[ b \approx 17 \times 0.8618 \approx 14.65 \text{ cm} \]
Rounding to the nearest centimeter:
\[ b \approx 15 \text{ cm} \]
Therefore, the length of side \(b\) is approximately \(\boxed{15}\) cm.
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