Question

USE The Law of Sines

TRIANGLE:
b: 15 in
a: 10 in
B: 85 degree
C:40 degree
Use the Law of Sines to find the value of side length c. Round to the nearest tenth of an inch

a. 0.2 inches
b. 6.5 inches
c. 9.7 inches
d. 15.5 inches

Answer 1

To use the Law of Sines to find side length \( c \) in triangle \( ABC \), we start by noting the given information:

- Side \( b = 15 \, \text{in} \)
- Side \( a = 10 \, \text{in} \)
- Angle \( B = 85^\circ \)
- Angle \( C = 40^\circ \)

First, we need to find angle \( A \) using the triangle angle sum property:

\[
A + B + C = 180^\circ
\]

Substituting the known angles:

\[
A + 85^\circ + 40^\circ = 180^\circ
\]

\[
A + 125^\circ = 180^\circ
\]

\[
A = 180^\circ - 125^\circ = 55^\circ
\]

Now that we have all angles, we apply the Law of Sines:

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

Using the known values, we can set up the equation for side \( c \):

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

Substituting \( A, a, C \):

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

Now, we can calculate \( c \):

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

Now, calculating the sine values:

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

Substituting these values:

\[
c = \frac{10 \cdot 0.6428}{0.8192} \approx \frac{6.428}{0.8192} \approx 7.85 \, \text{in}
\]

Rounding this to the nearest tenth gives:

\[
c \approx 7.9 \, \text{in}
\]

However, let's choose the closest available answer from the options given:

a. 0.2 inches
b. 6.5 inches
c. 9.7 inches
d. 15.5 inches

Since 7.9 inches is not one of the options, let's check the calculations quickly again:

- Using \( c = \frac{10 \cdot 0.6428}{0.8192} \)
- Checking the calculation of \(\frac{6.428}{0.8192}\) again yields approximately 7.85.

It seems the answer options do not match our calculated value of 7.9, but since 9.7 would be closer as a rough estimate:

Thus, we can conclude that none of the provided options are correct based on our calculations.

However, the closest calculated value compared to given options suggests we can assume either recalculation or alternative input for values that yield a reasonable \( c \approx 9.7 \).

So if pressed with the choices, I would indicate c. 9.7 inches as the proximate match.