Question

USE The Law of Sines

Given angle B is 94 degrees, use the law of sines to find the measure of angle C, if b = 13 inches and c = 6 inches. Round your answer to the nearest degree.

a. 7 degrees
b. 27 degrees
c. 86 degrees
d. 1 degrees

Answer 1

To find the measure of angle \( C \) using the Law of Sines, we start with the formula:

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

From the problem, we know:

- \( B = 94^\circ \)
- \( b = 13 \) inches
- \( c = 6 \) inches

We can express the ratios involving \( \sin C \):

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

Substituting what we know:

\[
\frac{6}{\sin C} = \frac{13}{\sin(94^\circ)}
\]

Now we need to calculate \(\sin(94^\circ)\). Since \(94^\circ\) is close to \(90^\circ\), we can use a calculator to find:

\[
\sin(94^\circ) \approx 0.998
\]

Next, substituting this value into our equation gives us:

\[
\frac{6}{\sin C} = \frac{13}{0.998}
\]

Cross-multiplying gives:

\[
6 \cdot 0.998 = 13 \cdot \sin C
\]

Calculating the left side:

\[
5.988 \approx 13 \cdot \sin C
\]

Now, solving for \(\sin C\):

\[
\sin C = \frac{5.988}{13} \approx 0.461
\]

Now, use the arcsine function to find \( C \):

\[
C = \arcsin(0.461)
\]

Using a calculator to find this:

\[
C \approx 27.5^\circ
\]

Rounding to the nearest degree:

\[
C \approx 28^\circ
\]

Based on the options provided, the closest option is:

**b. 27 degrees**.