Question

If in oblique ∆ ABC side a = 13 m, side b = 6 m, and side c = 9 m, the use the Law of Cosines to find the measure of ∠ A.

a. 15 degrees
b. 63 degrees
c. 119 degrees
d. 50 degrees

Answer 1

To find the measure of angle \( A \) using the Law of Cosines, we can use the formula:

\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
\]

Given:
- \( a = 13 \, m \)
- \( b = 6 \, m \)
- \( c = 9 \, m \)

Plugging in the values:

\[
\cos A = \frac{6^2 + 9^2 - 13^2}{2 \cdot 6 \cdot 9}
\]

Calculating each term:

\[
6^2 = 36
\]
\[
9^2 = 81
\]
\[
13^2 = 169
\]

Now compute \( b^2 + c^2 - a^2 \):

\[
36 + 81 - 169 = 117 - 169 = -52
\]

Now calculate \( 2bc \):

\[
2 \cdot 6 \cdot 9 = 108
\]

Substituting back into the cosine formula:

\[
\cos A = \frac{-52}{108}
\]

Now simplify the fraction:

\[
\cos A = -\frac{26}{54} = -\frac{13}{27}
\]

Now, calculate \( A \) using the inverse cosine (arccos):

\[
A = \cos^{-1}\left(-\frac{13}{27}\right)
\]

Using a calculator to find the angle:

\[
A \approx 119 \text{ degrees}
\]

Thus, the measure of angle \( A \) is approximately:

\[
\boxed{119 \text{ degrees}}
\]