To find the angle that correctly completes the law of cosines equation for the given triangle, we start with the given side lengths:
[tex]\( a = 24 \)[/tex]
[tex]\( b = 25 \)[/tex]
[tex]\( c = 7 \)[/tex]
The law of cosines states:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\][/tex]
Substituting the given values:
[tex]\[
7^2 = 24^2 + 25^2 - 2(24)(25) \cos(C)
\][/tex]
Simplifying:
[tex]\[
49 = 576 + 625 - 2(24)(25) \cos(C)
\][/tex]
Combining the constants:
[tex]\[
49 = 1201 - 1200 \cos(C)
\][/tex]
Rearranging to solve for [tex]\( \cos(C) \)[/tex]:
[tex]\[
1200 \cos(C) = 1201 - 49
\][/tex]
[tex]\[
1200 \cos(C) = 1152
\][/tex]
[tex]\[
\cos(C) = \frac{1152}{1200}
\][/tex]
[tex]\[
\cos(C) = 0.96
\][/tex]
Now, we need to find the angle [tex]\(C\)[/tex] whose cosine is 0.96. We use the inverse cosine function:
[tex]\[
C = \arccos(0.96)
\][/tex]
Calculating the angle in degrees:
[tex]\[
C \approx 16.26^\circ
\][/tex]
So, the angle that completes the law of cosines for this triangle is approximately [tex]\(16.26^\circ\)[/tex].
