Question

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.

Answer 1

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

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

We know that:

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

We can first find the side length \( a \) using the formula:

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

This allows us to isolate the sine of angle \( C \):

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

Rearranging gives:

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

Plugging in the values we have:

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

Now, we calculate \( \sin(94^\circ) \):

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

Now substituting this value in:

\[
\sin C \approx \frac{6 \cdot 0.998}{13}
\]

Calculating this gives:

\[
\sin C \approx \frac{5.988}{13} \approx 0.4606
\]

Next, we find angle \( C \) by taking the inverse sine:

\[
C \approx \sin^{-1}(0.4606)
\]

Using a calculator, we find:

\[
C \approx 27.4^\circ
\]

Rounding this to the nearest degree gives:

\[
C \approx 27^\circ
\]

Therefore, the measure of angle \( C \) is approximately **27 degrees**.