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:
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Calculate \( b^2 \), \( c^2 \), and \( a^2 \):
- \( b^2 = 6^2 = 36 \)
- \( c^2 = 9^2 = 81 \)
- \( a^2 = 13^2 = 169 \)
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Substitute these values into the Law of Cosines equation:
\[ \cos(A) = \frac{36 + 81 - 169}{2 \times 6 \times 9} \] \[ \cos(A) = \frac{117 - 169}{108} \] \[ \cos(A) = \frac{-52}{108} \] \[ \cos(A) = \frac{-26}{54} = \frac{-13}{27} \]
- 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 \( 127.3° \).
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