Question

Draw the picture to help you solve this problem.

In Triangle ABC, the measure of Angle A = 45 degrees, the measure of Angle B = 65 degrees and a = 15.05. Use the law of sines to find b Round your answer to the nearest tenth.

Answer 1

To solve for side \( b \) in triangle \( ABC \) using the Law of Sines, we first need the relevant angles and sides. We know:

- \( \angle A = 45^\circ \)
- \( \angle B = 65^\circ \)
- \( a = 15.05 \)

First, we can find angle \( C \) using the fact that the sum of the angles in a triangle is \( 180^\circ \):

\[
\angle C = 180^\circ - \angle A - \angle B
\]
\[
\angle C = 180^\circ - 45^\circ - 65^\circ = 70^\circ
\]

Now we can apply the Law of Sines, which states:

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

Substituting our known values:

\[
\frac{15.05}{\sin 45^\circ} = \frac{b}{\sin 65^\circ}
\]

Next, we calculate \( \sin 45^\circ \) and \( \sin 65^\circ \):

\[
\sin 45^\circ = \frac{\sqrt{2}}{2} \approx 0.7071
\]
\[
\sin 65^\circ \approx 0.9063
\]

Now we can plug these values into our equation:

\[
\frac{15.05}{0.7071} = \frac{b}{0.9063}
\]

Simplifying the left-hand side:

\[
15.05 \div 0.7071 \approx 21.24
\]

Now we have:

\[
21.24 = \frac{b}{0.9063}
\]

To find \( b \), we solve for \( b \):

\[
b = 21.24 \times 0.9063 \approx 19.29
\]

Rounding to the nearest tenth, we find:

\[
b \approx 19.3
\]

Thus, the length of side \( b \) is approximately **19.3**.