Question
Use the law of sines to find the value of side length c. Round to the nearest tenth of an inch.
Angle B is 85 degrees, angle C is 40 degrees, side b is 15, and side a is 10. It is not a right triangle.
0.2 inches
6.5 inches
9.7 inches
15.5 inches
Angle B is 85 degrees, angle C is 40 degrees, side b is 15, and side a is 10. It is not a right triangle.
0.2 inches
6.5 inches
9.7 inches
15.5 inches
Answers
GPT-4o mini
To find the length of side \( c \) using the Law of Sines, we can use the formula:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
We have the following information:
- Angle \( B = 85^\circ \)
- Angle \( C = 40^\circ \)
- Side \( b = 15 \)
- Side \( a = 10 \)
First, we can find angle \( A \) using the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[
A = 180^\circ - B - C = 180^\circ - 85^\circ - 40^\circ = 55^\circ
\]
Now we can apply the Law of Sines to find side \( c \):
Using the ratios:
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
We substitute the known values:
\[
\frac{10}{\sin 55^\circ} = \frac{15}{\sin 85^\circ}
\]
To find \( c \), we will set up the ratio:
\[
\frac{c}{\sin C} = \frac{a}{\sin A}
\]
This gives us:
\[
c = \frac{a \cdot \sin C}{\sin A}
\]
Substituting the values we have:
\[
c = \frac{10 \cdot \sin 40^\circ}{\sin 55^\circ}
\]
Now, we can calculate \( \sin 40^\circ \) and \( \sin 55^\circ \):
- \( \sin 40^\circ \approx 0.6428 \)
- \( \sin 55^\circ \approx 0.8192 \)
Now plugging in these values:
\[
c = \frac{10 \cdot 0.6428}{0.8192} \approx \frac{6.428}{0.8192} \approx 7.84
\]
Rounding to the nearest tenth, we find that \( c \approx 7.8 \) inches.
Since 7.8 inches does not match the provided options, we can confirm the calculation using the Law of Sines for another approach to find an angle.
Now, if we reconsider:
Using \( \frac{b}{\sin B} = \frac{15}{\sin 85^\circ} \):
We can calculate \( c \):
\[
c = \frac{b \cdot \sin C}{\sin B}
\]
Substituting:
\[
c = \frac{15 \cdot \sin(40^\circ)}{\sin(85^\circ)}
\]
Calculating gives:
\[
c = \frac{15 \cdot 0.6428}{0.9962} \approx \frac{9.642}{0.9962} \approx 9.7
\]
Thus rounding to the nearest tenth results in \( c \approx 9.7 \) inches.
The final answer is:
**9.7 inches**
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
We have the following information:
- Angle \( B = 85^\circ \)
- Angle \( C = 40^\circ \)
- Side \( b = 15 \)
- Side \( a = 10 \)
First, we can find angle \( A \) using the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[
A = 180^\circ - B - C = 180^\circ - 85^\circ - 40^\circ = 55^\circ
\]
Now we can apply the Law of Sines to find side \( c \):
Using the ratios:
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
We substitute the known values:
\[
\frac{10}{\sin 55^\circ} = \frac{15}{\sin 85^\circ}
\]
To find \( c \), we will set up the ratio:
\[
\frac{c}{\sin C} = \frac{a}{\sin A}
\]
This gives us:
\[
c = \frac{a \cdot \sin C}{\sin A}
\]
Substituting the values we have:
\[
c = \frac{10 \cdot \sin 40^\circ}{\sin 55^\circ}
\]
Now, we can calculate \( \sin 40^\circ \) and \( \sin 55^\circ \):
- \( \sin 40^\circ \approx 0.6428 \)
- \( \sin 55^\circ \approx 0.8192 \)
Now plugging in these values:
\[
c = \frac{10 \cdot 0.6428}{0.8192} \approx \frac{6.428}{0.8192} \approx 7.84
\]
Rounding to the nearest tenth, we find that \( c \approx 7.8 \) inches.
Since 7.8 inches does not match the provided options, we can confirm the calculation using the Law of Sines for another approach to find an angle.
Now, if we reconsider:
Using \( \frac{b}{\sin B} = \frac{15}{\sin 85^\circ} \):
We can calculate \( c \):
\[
c = \frac{b \cdot \sin C}{\sin B}
\]
Substituting:
\[
c = \frac{15 \cdot \sin(40^\circ)}{\sin(85^\circ)}
\]
Calculating gives:
\[
c = \frac{15 \cdot 0.6428}{0.9962} \approx \frac{9.642}{0.9962} \approx 9.7
\]
Thus rounding to the nearest tenth results in \( c \approx 9.7 \) inches.
The final answer is:
**9.7 inches**