Question

a triangle has side lengths 4, 7 and 9. what is the measure of the angle across from the longest side?
$9^{2}=4^{2}+7^{2}-2(4)(7)\cos (a)$
$81 = 16 + 49 - 56\cos (a)$
$81 = 9\cos (a)$
$9 = \cos (a)$
a cannot exist!
gabe tried to use the law of cosines to find an unknown angle measure in a triangle. his work is shown. what is gabes error?
gabe reversed the order of the 9 and the 4.
gabe squared the numbers incorrectly.
gabe should not have subtracted 56 from $16 + 49$.
gabe incorrectly stated that $\cos^{-1}(9)$ is not defined.
done

Explanation:

Step1: Analyze the law of cosines formula

The law of cosines is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\), where \(c\) is the side opposite angle \(C\), and \(a\) and \(b\) are the other two sides. In the given problem, when using \(9^{2}=4^{2}+7^{2}-2(4)(7)\cos(A)\), the next step should be \(81 = 16+49 - 56\cos(A)\) (correct squaring: \(4^{2}=16\), \(7^{2} = 49\), \(9^{2}=81\)).

Step2: Check the arithmetic

However, \(16 + 49=65\), not \(9\). The correct equation after substitution is \(81=65 - 56\cos(A)\), not \(81 = 9\cos(A)\). Gabe should not have subtracted \(56\) from \(16 + 49\) (he made an arithmetic error in adding \(16\) and \(49\)).

Answer:

Gabe should not have subtracted 56 from \(16 + 49\).