Question

The length of side c is 21 inches. If angle C is 47 degrees and angle A is 22 degrees, use the Law of Sines to find the length of side a. Round your answer to the nearest tenth of an inch. (1 point) inches

Answer 1

To find the length of side \( a \) using the Law of Sines, we can use the formula:

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

Where:
- \( A \) and \( C \) are the angles,
- \( a \) and \( c \) are the lengths of the sides opposite those angles.

Given:
- \( c = 21 \) inches
- \( C = 47^\circ \)
- \( A = 22^\circ \)

First, we need to calculate the measure of angle \( B \):

\[
B = 180^\circ - A - C = 180^\circ - 22^\circ - 47^\circ = 111^\circ
\]

Now we can use the Law of Sines to find \( a \):

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

Plugging in the known values:

\[
\frac{a}{\sin(22^\circ)} = \frac{21}{\sin(47^\circ)}
\]

Now we can solve for \( a \):

\[
a = \frac{21 \cdot \sin(22^\circ)}{\sin(47^\circ)}
\]

Calculating the values of the sine functions:

\[
\sin(22^\circ) \approx 0.3746
\]
\[
\sin(47^\circ) \approx 0.7314
\]

Substituting these values into the equation:

\[
a \approx \frac{21 \cdot 0.3746}{0.7314}
\]
\[
a \approx \frac{7.8636}{0.7314} \approx 10.75
\]

Rounding to the nearest tenth:

\[
a \approx 10.8 \text{ inches}
\]

Thus, the length of side \( a \) is approximately **10.8 inches**.