To determine the angle that completes the law of cosines for the given triangle, we will use the law of cosines formula:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \][/tex]
Given:
- [tex]\( a = 12 \)[/tex]
- [tex]\( b = 13 \)[/tex]
- [tex]\( c = 5 \)[/tex]
Let's plug these values into the law of cosines formula to solve for [tex]\( \cos(C) \)[/tex]:
[tex]\[ 5^2 = 12^2 + 13^2 - 2 \cdot 12 \cdot 13 \cdot \cos(C) \][/tex]
Simplifying, we get:
[tex]\[ 25 = 144 + 169 - 312 \cdot \cos(C) \][/tex]
Combining the constants:
[tex]\[ 25 = 313 - 312 \cdot \cos(C) \][/tex]
Rearrange the equation to solve for [tex]\( \cos(C) \)[/tex]:
[tex]\[ 312 \cdot \cos(C) = 313 - 25 \][/tex]
[tex]\[ 312 \cdot \cos(C) = 288 \][/tex]
Next, isolate [tex]\( \cos(C) \)[/tex]:
[tex]\[ \cos(C) = \frac{288}{312} \][/tex]
[tex]\[ \cos(C) = \frac{24}{26} \][/tex]
[tex]\[ \cos(C) = \frac{12}{13} \][/tex]
Now, we need to find the angle [tex]\( C \)[/tex] such that [tex]\( \cos(C) = \frac{12}{13} \)[/tex]. Using the inverse cosine function ([tex]\( \cos^{-1} \)[/tex]), we find:
[tex]\[ C = \cos^{-1} \left(\frac{12}{13}\right) \][/tex]
The value of this angle, [tex]\( C \)[/tex], is calculated to be approximately [tex]\( 22.61986494804042^{\circ} \)[/tex].
Given the multiple-choice options:
A. [tex]\( 23^{\circ} \)[/tex]
B. [tex]\( 90^{\circ} \)[/tex]
C. [tex]\( 180^{\circ} \)[/tex]
D. [tex]\( 67^{\circ} \)[/tex]
The angle closest to our calculated value of [tex]\( 22.61986494804042^{\circ} \)[/tex] is:
A. [tex]\( 23^{\circ} \)[/tex]
Thus, the angle that correctly completes the law of cosines for this triangle is:
[tex]\[ \boxed{23^{\circ}} \][/tex]
