Question

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

Answer 1

To find the angle ∠A in triangle ABC using the Law of Cosines, we use the formula:

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

where \( a \), \( b \), and \( c \) are the lengths of the sides opposite to angles A, B, and C, respectively. In this case:

- \( a = 13 \) m (opposite angle A)
- \( b = 6 \) m (opposite angle B)
- \( c = 9 \) m (opposite angle C)

We can substitute the values into the formula:

1. Calculate \( b^2 \), \( c^2 \), and \( a^2 \):
- \( b^2 = 6^2 = 36 \)
- \( c^2 = 9^2 = 81 \)
- \( a^2 = 13^2 = 169 \)

2. Substitute these values into the Law of Cosines equation:

\[
\cos(A) = \frac{36 + 81 - 169}{2 \times 6 \times 9}
\]
\[
\cos(A) = \frac{117 - 169}{108}
\]
\[
\cos(A) = \frac{-52}{108}
\]
\[
\cos(A) = \frac{-26}{54} = \frac{-13}{27}
\]

3. Now, we find the angle A using the arccos function:
\[
A = \cos^{-1}\left(\frac{-13}{27}\right)
\]

Using a calculator, we can compute this value:

\[
A \approx 127.29° \text{ (approximately)}
\]

Thus, the measure of angle A is approximately \( 127.3° \).