Question

For ΔABC , find the measure of ∠C to the nearest degree when side a=46 m, side b=30 m, and side c=20 m.

Answer 1

To find the measure of angle \( C \) in triangle \( ABC \) given the sides \( a \), \( b \), and \( c \), we can use the Law of Cosines, which states:

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

Where:
- \( a = 46 \) m
- \( b = 30 \) m
- \( c = 20 \) m
- \( C \) is the angle opposite side \( c \).

First, we rearrange the formula to solve for \( \cos(C) \):

\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]

Now, we substitute the given values into the formula:

\[
a^2 = 46^2 = 2116
\]
\[
b^2 = 30^2 = 900
\]
\[
c^2 = 20^2 = 400
\]

Now, plug these into the formula for \( \cos(C) \):

\[
\cos(C) = \frac{2116 + 900 - 400}{2 \times 46 \times 30}
\]

Calculating the numerator:

\[
2116 + 900 - 400 = 2616
\]

Calculating the denominator:

\[
2 \times 46 \times 30 = 2760
\]

Now, we find \( \cos(C) \):

\[
\cos(C) = \frac{2616}{2760}
\]

Now, we simplify:

\[
\cos(C) \approx 0.948
\]

Now we find angle \( C \) by taking the inverse cosine:

\[
C = \cos^{-1}(0.948)
\]

Using a calculator:

\[
C \approx 18.19^\circ
\]

Rounding to the nearest degree:

\[
C \approx 18^\circ
\]

Thus, the measure of \( \angle C \) is approximately \( \boxed{18}^\circ \).