Question
use the law of sines to solve the following problem. if <c= 62 degrees and side c= 16 cm, then what is the length of side a? round your answer to the nearest centimeter
Answers
GPT-4o mini
To use the Law of Sines to find the length of side \( a \), we need to know the lengths of side \( c \) and the angle \( C \), along with either angle \( A \) or angle \( B \) in the triangle.
The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Given:
- \( C = 62^\circ \)
- \( c = 16 \, \text{cm} \)
To find side \( a \), we need one more angle. However, since we only have this data, I will assume that we still need angle \( A \) or \( B \) to find \( a \).
If we assume we know one more angle \( A \) (for example, let's say \( A = 30^\circ \)), we can apply the Law of Sines. But since I do not have that information, I'll show you how to solve it with an assumption.
### Example Calculation (Assuming \( A = 30^\circ \))
1. Calculate angle \( B \) using the triangle angle sum property:
\[
A + B + C = 180^\circ
\]
\[
30^\circ + B + 62^\circ = 180^\circ
\]
\[
B = 180^\circ - 30^\circ - 62^\circ = 88^\circ
\]
2. Now, apply the Law of Sines:
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Substituting the values we have:
\[
\frac{a}{\sin 30^\circ} = \frac{16}{\sin 62^\circ}
\]
Where \( \sin 30^\circ = 0.5 \) and \( \sin 62^\circ \approx 0.8746 \).
3. Rearranging the equation to solve for \( a \):
\[
a = \frac{16 \cdot \sin 30^\circ}{\sin 62^\circ}
\]
\[
a = \frac{16 \cdot 0.5}{0.8746} \approx \frac{8}{0.8746} \approx 9.14 \, \text{cm}
\]
4. Rounding to the nearest centimeter:
\[
a \approx 9 \, \text{cm}
\]
### Final Note:
You will need the actual value of angle \( A \) or \( B \) to perform a specific calculation. If you have that information, follow the same process to solve for side \( a \).
The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Given:
- \( C = 62^\circ \)
- \( c = 16 \, \text{cm} \)
To find side \( a \), we need one more angle. However, since we only have this data, I will assume that we still need angle \( A \) or \( B \) to find \( a \).
If we assume we know one more angle \( A \) (for example, let's say \( A = 30^\circ \)), we can apply the Law of Sines. But since I do not have that information, I'll show you how to solve it with an assumption.
### Example Calculation (Assuming \( A = 30^\circ \))
1. Calculate angle \( B \) using the triangle angle sum property:
\[
A + B + C = 180^\circ
\]
\[
30^\circ + B + 62^\circ = 180^\circ
\]
\[
B = 180^\circ - 30^\circ - 62^\circ = 88^\circ
\]
2. Now, apply the Law of Sines:
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Substituting the values we have:
\[
\frac{a}{\sin 30^\circ} = \frac{16}{\sin 62^\circ}
\]
Where \( \sin 30^\circ = 0.5 \) and \( \sin 62^\circ \approx 0.8746 \).
3. Rearranging the equation to solve for \( a \):
\[
a = \frac{16 \cdot \sin 30^\circ}{\sin 62^\circ}
\]
\[
a = \frac{16 \cdot 0.5}{0.8746} \approx \frac{8}{0.8746} \approx 9.14 \, \text{cm}
\]
4. Rounding to the nearest centimeter:
\[
a \approx 9 \, \text{cm}
\]
### Final Note:
You will need the actual value of angle \( A \) or \( B \) to perform a specific calculation. If you have that information, follow the same process to solve for side \( a \).