To determine which angle correctly completes the law of cosines for the given triangle, we start with the given equation:
[tex]\[ 7^2 + 25^2 - 2(7)(25) \cos C = 24^2 \][/tex]
First, simplify the equation step-by-step to solve for \(\cos C\).
1. Compute the squares of each side:
[tex]\[ 7^2 = 49 \][/tex]
[tex]\[ 25^2 = 625 \][/tex]
[tex]\[ 24^2 = 576 \][/tex]
2. Substitute these values into the equation:
[tex]\[ 49 + 625 - 2(7)(25) \cos C = 576 \][/tex]
3. Simplify the left side:
[tex]\[ 674 - 2(7)(25) \cos C = 576 \][/tex]
4. Calculate the product:
[tex]\[ 2 \cdot 7 \cdot 25 = 350 \][/tex]
5. Substitute back into the equation:
[tex]\[ 674 - 350 \cos C = 576 \][/tex]
6. Isolate the term involving \(\cos C\):
[tex]\[ 674 - 576 = 350 \cos C \][/tex]
[tex]\[ 98 = 350 \cos C \][/tex]
7. Solve for \(\cos C\):
[tex]\[ \cos C = \frac{98}{350} \][/tex]
[tex]\[ \cos C = \frac{98}{350} = 0.28 \][/tex]
Next, we need to find the angle \(C\) whose cosine value is \(0.28\). This can be done by taking the arccosine (inverse cosine):
[tex]\[ C = \cos^{-1}(0.28) \][/tex]
Computing this value gives us:
[tex]\[ C \approx 73.74^\circ \][/tex]
Now, compare this value to the given options:
A. \(74^\circ\)
B. \(16^\circ\)
C. \(180^\circ\)
D. \(90^\circ\)
The closest angle to \(73.74^\circ\) is \(74^\circ\).
Therefore, the correct angle that completes the law of cosines for this triangle is:
A. [tex]\(74^\circ\)[/tex]
