To solve for the length of side [tex]\( c \)[/tex] in a triangle given two sides [tex]\( a = 5 \)[/tex], [tex]\( b = 12 \)[/tex], and the included angle [tex]\( C = 72^\circ \)[/tex], we can use the Law of Cosines. The Law of Cosines formula states that for any triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], and the included angle [tex]\( C \)[/tex], the following equation holds:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\][/tex]
Now we can substitute the given values into the formula:
[tex]\[
c^2 = 5^2 + 12^2 - 2 \cdot 5 \cdot 12 \cos(72^\circ)
\][/tex]
Let’s break this down step by step:
1. Square the sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 12^2 = 144 \][/tex]
2. Multiply the sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex], then multiply by 2:
[tex]\[ 2 \cdot 5 \cdot 12 = 120 \][/tex]
3. Multiply by the cosine of the included angle:
[tex]\[ 120 \cos(72^\circ) \][/tex]
4. Substitute these values back into the equation:
[tex]\[ c^2 = 25 + 144 - 120 \cos(72^\circ) \][/tex]
So, the equation that can be used to solve for the length of side [tex]\( c \)[/tex] is:
[tex]\[
c^2 = 5^2 + 12^2 - 2(5)(12) \cos 72^{\circ}
\][/tex]
Now let's match this with the given options:
1. [tex]\( 2^2-5^3+12^2-2(5)(12) \cos 72^{\circ} \)[/tex]
2. [tex]\( c^2-5^2+12^2+2(5)(12) \cos 72^{\circ} \)[/tex]
3. [tex]\( c^2-5^2-12^3-2(5)(12) \cos 72^{\circ} \)[/tex]
4. [tex]\( 2^2-5^2-12^2+2(5)(12) \cos 72^{\circ} \)[/tex]
Clearly, none of these precisely match our equation directly. However, let's analyze closely and let us correct the available choices. Given the closest 'correct' form is a common sense tool try to actively switch as:
- The third choice forces a positive + in [tex]\(\cos 72\)[/tex] invalidating cosine's structure in the second term.
- Second choice also from both syntactic thus basic math law, i.e., Order the cosine and keep sign convention, plainly invalid too.
Thus reconsider the suitable corrections aligning in boundary maths but correct,
[tex]\[
c^2 = 5^2 + 12^2 - 2(5)(12) \cos 72^{\circ}
\][/tex]
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