Select the angle that correctly completes the law of cosines for this triangle.

[tex]\[ 12^2 + 13^2 - 2(12)(13) \cos \theta = 5^2 \][/tex]

A. \( 180^{\circ} \)
B. \( 90^{\circ} \)
C. \( 23^{\circ} \)
D. [tex]\( 67^{\circ} \)[/tex]



Answer :

Let's explore the problem step-by-step using the Law of Cosines:

The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]

Given:
[tex]\[ a = 12 \][/tex]
[tex]\[ b = 13 \][/tex]
[tex]\[ c = 5 \][/tex]

We need to find the angle \( C \). According to the Law of Cosines:

[tex]\[ 5^2 = 12^2 + 13^2 - 2 \cdot 12 \cdot 13 \cdot \cos(C) \][/tex]

First, let's plug in the values:

[tex]\[ 25 = 144 + 169 - 312 \cos(C) \][/tex]

Combine the values:

[tex]\[ 25 = 313 - 312 \cos(C) \][/tex]

Rearrange to solve for \( \cos(C) \):

[tex]\[ 312 \cos(C) = 313 - 25 \][/tex]
[tex]\[ 312 \cos(C) = 288 \][/tex]
[tex]\[ \cos(C) = \frac{288}{312} \][/tex]
[tex]\[ \cos(C) = \frac{24}{26} \][/tex]
[tex]\[ \cos(C) = \frac{12}{13} \][/tex]

Now, to find the angle \( C \), we take the arccos (inverse cosine) of \(\frac{12}{13}\):

[tex]\[ C = \arccos \left(\frac{12}{13}\right) \][/tex]

Using a calculator or tables, we find:

[tex]\[ C \approx 22.62^{\circ} \][/tex]

Among the given choices, the angle that fits closest is:

C. \( 23^{\circ} \)

Therefore, the correct answer is [tex]\( 23^{\circ} \)[/tex].