To find the length of side \( b \) in triangle \( ABC \) using the Law of Sines, we can start by determining the remaining angle \( C \).
The sum of the angles in a triangle is \( 180^\circ \), so:
\[ C = 180^\circ - A - B \] \[ C = 180^\circ - 14^\circ - 52^\circ = 114^\circ \]
Now we can use the Law of Sines, which states:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
We have \( c = 17 \) cm, \( A = 14^\circ \), and \( B = 52^\circ \). We want to find \( b \).
Using the Law of Sines, we set up the equation:
\[ \frac{b}{\sin B} = \frac{c}{\sin C} \]
Substituting the known values:
\[ \frac{b}{\sin 52^\circ} = \frac{17}{\sin 114^\circ} \]
Using a calculator to find the sine values:
\[ \sin 52^\circ \approx 0.7880 \quad \text{and} \quad \sin 114^\circ \approx 0.9135 \]
Now we can substitute these values into the equation:
\[ \frac{b}{0.7880} = \frac{17}{0.9135} \]
Cross-multiplying gives:
\[ b \cdot 0.9135 = 17 \cdot 0.7880 \]
Calculating the right side:
\[ 17 \cdot 0.7880 \approx 13.396 \]
Now we can solve for \( b \):
\[ b = \frac{13.396}{0.9135} \approx 14.7 \]
Rounding this to the nearest centimeter, we get:
\[ b \approx 15 \text{ cm} \]
Thus, the length of side \( b \) is approximately 15 cm.
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