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.

Answer 1

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

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

We know the following:

- Length of side \( c = 21 \) inches
- Angle \( C = 47^\circ \)
- Angle \( A = 22^\circ \)

First, we need to determine angle \( B \) using the fact that the sum of angles in a triangle equals \( 180^\circ \):

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

Now we can substitute the known values into the Law of Sines equation:

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

Next, we can solve for \( a \):

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

Now, we need the sine values:

- \( \sin(22^\circ) \approx 0.3746 \)
- \( \sin(47^\circ) \approx 0.7314 \)

Now substitute these values into the equation:

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

Calculate \( a \):

\[
a \approx \frac{7.8626}{0.7314} \approx 10.75
\]

Rounding to the nearest tenth gives:

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

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