Question

In the triangle, angle B is 52 degrees and angle A  is 14 degrees. If side c has a length of 17 cm, use the Law of Sines to find the length of side b. Round your answer to the nearest centimeter.

Answer 1

To find the length of side \( b \) in the triangle using the Law of Sines, we start by determining the missing angle \( C \).

The sum of the angles in a triangle is \( 180^\circ \):

\[
C = 180^\circ - A - B
\]
Given \( A = 14^\circ \) and \( B = 52^\circ \):

\[
C = 180^\circ - 14^\circ - 52^\circ = 114^\circ
\]

Now, we can use the Law of Sines, which states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

We need to find the length of \( b \). Using \( c = 17 \, \text{cm} \) and \( \angle C = 114^\circ \):

\[
\frac{b}{\sin B} = \frac{c}{\sin C}
\]

Substituting the known values:

\[
\frac{b}{\sin 52^\circ} = \frac{17}{\sin 114^\circ}
\]

Now we calculate \( \sin 52^\circ \) and \( \sin 114^\circ \):

- \( \sin 52^\circ \approx 0.7880 \)
- \( \sin 114^\circ \approx 0.9117 \)

Substituting these values into the equation gives:

\[
\frac{b}{0.7880} = \frac{17}{0.9117}
\]

Now, we can solve for \( b \):

\[
b = 17 \cdot \frac{0.7880}{0.9117}
\]

Calculating the right side:

\[
b \approx 17 \cdot 0.8655 \approx 14.69
\]

Rounding to the nearest centimeter, the length of side \( b \) is:

\[
\boxed{15} \, \text{cm}
\]