Question

use the law of sines to find the value of side length c. round to the nearest tenth of an inch.

/_C = 40 degrees
b = 15 inches
a = 10 inches
/_B = 85 degrees

Answer 1

To find the length of side \( c \) using the Law of Sines, we can start with the formula:

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

From your data, we have:
- \( a = 10 \) inches
- \( b = 15 \) inches
- \( \angle C = 40^\circ \)
- \( \angle B = 85^\circ \)

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

\[
\angle A = 180^\circ - \angle B - \angle C
\]
\[
\angle A = 180^\circ - 85^\circ - 40^\circ = 55^\circ
\]

Now we can apply the Law of Sines to find \( c \):

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

Substituting the known values:

\[
\frac{10}{\sin 55^\circ} = \frac{c}{\sin 40^\circ}
\]

Cross-multiplying gives us:

\[
10 \cdot \sin 40^\circ = c \cdot \sin 55^\circ
\]

Now we can solve for \( c \):

\[
c = \frac{10 \cdot \sin 40^\circ}{\sin 55^\circ}
\]

Now calculate \( \sin 40^\circ \) and \( \sin 55^\circ \):

- \( \sin 40^\circ \approx 0.6428 \)
- \( \sin 55^\circ \approx 0.8192 \)

Substitute these values into the equation for \( c \):

\[
c \approx \frac{10 \cdot 0.6428}{0.8192}
\]
\[
c \approx \frac{6.428}{0.8192} \approx 7.846
\]

Rounding to the nearest tenth, we get:

\[
c \approx 7.8 \text{ inches}
\]

Thus, the value of side length \( c \) is approximately **7.8 inches**.