Question

USE THE LAW OF SINES

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.

___ 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:
- \( a \) is the side opposite angle \( A \),
- \( c \) is the side opposite angle \( C \),
- \( A \) and \( C \) are the angles given in degrees.

From the problem, we have:
- \( c = 21 \) inches
- \( C = 47^\circ \)
- \( A = 22^\circ \)

First, we need to calculate \( \sin A \) and \( \sin C \):

- \( \sin A = \sin(22^\circ) \)
- \( \sin C = \sin(47^\circ) \)

Now, using a calculator:

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

Now we can set up the equation according to the Law of Sines:

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

Substituting the values into the equation:

\[
\frac{a}{0.3746} = \frac{21}{0.7314}
\]

First, calculate the right side:

\[
\frac{21}{0.7314} \approx 28.7
\]

Now the equation is:

\[
\frac{a}{0.3746} = 28.7
\]

To solve for \( a \), multiply both sides by \( 0.3746 \):

\[
a = 28.7 \times 0.3746 \approx 10.75
\]

Rounding to the nearest tenth, we find:

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

Thus, the length of side \( a \) is approximately \( \boxed{10.8} \) inches.