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.
(1 point)
cm
and side c=16 cm
, then what is the length of side a? Round your answer to the nearest centimeter.
(1 point)
cm
Answers
GPT-4o mini
To find the length of side \( a \) using the Law of Sines, we start with the formula:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Where:
- \( a \) is the side opposite angle \( A \),
- \( b \) is the side opposite angle \( B \),
- \( c \) is the side opposite angle \( C \),
- \( \angle C = 62^\circ \),
- \( c = 16 \) cm.
However, we need either angle \( A \) or angle \( B \) to find side \( a \), so let's assume we have a triangle where the total interior angles sum to \( 180^\circ \):
\[
A + B + C = 180^\circ
\]
Without loss of generality, let's say we know angle \( A \) or angle \( B \).
For example, if we assume angle \( A \) is \( 30^\circ \) (as an arbitrary choice for illustration), we can first calculate angle \( B \):
\[
B = 180^\circ - A - C = 180^\circ - 30^\circ - 62^\circ = 88^\circ
\]
Now we have:
- \( C = 62^\circ \)
- \( c = 16 \) cm
- \( A = 30^\circ \)
- \( B = 88^\circ \)
Now we can use the Law of Sines to find \( a \):
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Substituting in the known values:
\[
\frac{a}{\sin 30^\circ} = \frac{16}{\sin 62^\circ}
\]
Calculating \( \sin 30^\circ = 0.5 \) and \( \sin 62^\circ \) (approximately \( 0.8746 \)):
\[
\frac{a}{0.5} = \frac{16}{0.8746}
\]
Cross-multiplying gives:
\[
a = 0.5 \cdot \frac{16}{0.8746}
\]
\[
a = 0.5 \cdot 18.26 \approx 9.13
\]
Rounded to the nearest centimeter, we find:
\[
a \approx 9 \text{ cm}
\]
However, this solution depends on the value of angle \( A \). The actual length of side \( a \) will vary based on the known angles in the triangle.
If you provide the values of \( A \) or \( B \), I can give you an exact solution for \( a \). Otherwise, here's an all-purpose answer given our working example: side \( a \approx 9 \) cm.
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Where:
- \( a \) is the side opposite angle \( A \),
- \( b \) is the side opposite angle \( B \),
- \( c \) is the side opposite angle \( C \),
- \( \angle C = 62^\circ \),
- \( c = 16 \) cm.
However, we need either angle \( A \) or angle \( B \) to find side \( a \), so let's assume we have a triangle where the total interior angles sum to \( 180^\circ \):
\[
A + B + C = 180^\circ
\]
Without loss of generality, let's say we know angle \( A \) or angle \( B \).
For example, if we assume angle \( A \) is \( 30^\circ \) (as an arbitrary choice for illustration), we can first calculate angle \( B \):
\[
B = 180^\circ - A - C = 180^\circ - 30^\circ - 62^\circ = 88^\circ
\]
Now we have:
- \( C = 62^\circ \)
- \( c = 16 \) cm
- \( A = 30^\circ \)
- \( B = 88^\circ \)
Now we can use the Law of Sines to find \( a \):
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Substituting in the known values:
\[
\frac{a}{\sin 30^\circ} = \frac{16}{\sin 62^\circ}
\]
Calculating \( \sin 30^\circ = 0.5 \) and \( \sin 62^\circ \) (approximately \( 0.8746 \)):
\[
\frac{a}{0.5} = \frac{16}{0.8746}
\]
Cross-multiplying gives:
\[
a = 0.5 \cdot \frac{16}{0.8746}
\]
\[
a = 0.5 \cdot 18.26 \approx 9.13
\]
Rounded to the nearest centimeter, we find:
\[
a \approx 9 \text{ cm}
\]
However, this solution depends on the value of angle \( A \). The actual length of side \( a \) will vary based on the known angles in the triangle.
If you provide the values of \( A \) or \( B \), I can give you an exact solution for \( a \). Otherwise, here's an all-purpose answer given our working example: side \( a \approx 9 \) cm.