Question

Imagine that a is 5 feet, b is 21 feet, and c is 18 feet. If angle B is 55 degrees, find angle C . Round your answer to the nearest degree.

Answer 1

To find angle \( C \), we can use the Law of Cosines first to find angle \( A \), and then we can find angle \( C \) using the fact that the sum of the angles in a triangle is \( 180^\circ \).

The Law of Cosines states:

\[
c^2 = a^2 + b^2 - 2ab \cos(B)
\]

Given:
- \( a = 5 \) feet
- \( b = 21 \) feet
- \( c = 18 \) feet
- \( B = 55^\circ \)

Let's plug in the values into the Law of Cosines formula to find \( c^2 \):

\[
18^2 = 5^2 + 21^2 - 2(5)(21) \cos(55^\circ)
\]

Calculating \( a^2 \), \( b^2 \), and \( c^2 \):

\[
18^2 = 324
\]

\[
5^2 = 25, \quad 21^2 = 441
\]

Now substitute these into the equation:

\[
324 = 25 + 441 - 2(5)(21) \cos(55^\circ)
\]

Calculating \( 25 + 441 \):

\[
324 = 466 - 210 \cos(55^\circ)
\]

Now isolate \( 210 \cos(55^\circ) \):

\[
210 \cos(55^\circ) = 466 - 324
\]

\[
210 \cos(55^\circ) = 142
\]

Now, solve for \( \cos(55^\circ) \):

\[
\cos(55^\circ) = \frac{142}{210}
\]

Calculating \( \frac{142}{210} \):

\[
\cos(55^\circ) \approx 0.67619
\]

Now we can use the Law of Sines to find angle \( C \):

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

First, we need to determine angle \( A \) using:

\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)}
\]

Rearranging gives us:

\[
\sin(A) = \frac{a \sin(B)}{b}
\]

Plugging in the values:

\[
\sin(A) = \frac{5 \sin(55^\circ)}{21}
\]

Calculating \( \sin(55^\circ) \):

\[
\sin(55^\circ) \approx 0.81915
\]

Thus:

\[
\sin(A) = \frac{5 \times 0.81915}{21} \approx \frac{4.09575}{21} \approx 0.195
\]

Now calculating angle \( A \):

\[
A \approx \sin^{-1}(0.195) \approx 11.3^\circ
\]

Now we can find angle \( C \):

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

Substituting the values we found:

\[
C = 180^\circ - 11.3^\circ - 55^\circ \approx 113.7^\circ
\]

Rounding \( C \) to the nearest degree:

\[
C \approx 114^\circ
\]

Thus, angle \( C \) is approximately \( \boxed{114} \) degrees.