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'll first need to find angle \( B \). Since the sum of the angles in a triangle is \( 180^\circ \), we can calculate angle \( B \) as follows:

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

Now that we have all the angles, we'll use the Law of Sines, which states:

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

From the information given:

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

We will set up the equation for side \( a \):

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

Substituting the known values:

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

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

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

Next, we need to calculate \( \sin(22^\circ) \) and \( \sin(47^\circ) \):

Using a calculator:

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

Now plug those values into the equation:

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

Calculating further:

\[
a \approx \frac{7.8736}{0.7314} \approx 10.77
\]

Rounding to the nearest tenth:

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

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