To find the length of side \( b \) in the triangle using the Law of Sines, we first need to determine the measures of all angles in the triangle. We know:
- \( A = 14^\circ \)
- \( B = 52^\circ \)
We can find the third angle \( C \) using the fact that the sum of angles in a triangle is \( 180^\circ \):
\[ C = 180^\circ - A - B = 180^\circ - 14^\circ - 52^\circ = 114^\circ \]
Now we can use the Law of Sines, which states:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Given that side \( c = 17 \) cm, we can find side \( b \):
\[ \frac{b}{\sin B} = \frac{c}{\sin C} \]
Substituting in the known values:
\[ \frac{b}{\sin 52^\circ} = \frac{17}{\sin 114^\circ} \]
First, we need the sine values:
- \( \sin 52^\circ \) (approx. 0.788)
- \( \sin 114^\circ = \sin (180^\circ - 66^\circ) = \sin 66^\circ \) (approx. 0.913)
Now, substituting these values into the equation:
\[ \frac{b}{0.788} = \frac{17}{0.913} \]
Cross-multiplying gives:
\[ b \cdot 0.913 = 17 \cdot 0.788 \]
Calculating the right side:
\[ b \cdot 0.913 = 13.396 \]
Now, solving for \( b \):
\[ b = \frac{13.396}{0.913} \approx 14.7 \]
Rounding to the nearest centimeter, we find:
\[ b \approx 15 \text{ cm} \]
Thus, the length of side \( b \) is approximately 15 cm.
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