To find the length of the third side of the triangle when the lengths of two sides and the included angle are given, we can use the Law of Cosines. The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the given sides, [tex]\( C \)[/tex] is the included angle, and [tex]\( c \)[/tex] is the length of the third side. Given:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( C = 60^{\circ} \)[/tex]
First, we convert the angle [tex]\( C \)[/tex] to radians (which necessitates knowing that [tex]\( 60^{\circ} \)[/tex] in radians is [tex]\(\frac{\pi}{3}\)[/tex]). The cosine of [tex]\( 60^{\circ} \)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
Now, applying the Law of Cosines:
[tex]\[ c^2 = 2^2 + 3^2 - 2 \cdot 2 \cdot 3 \cdot \cos(60^{\circ}) \][/tex]
We substitute the given values and simplify:
[tex]\[ c^2 = 4 + 9 - 2 \cdot 2 \cdot 3 \cdot \frac{1}{2} \][/tex]
[tex]\[ c^2 = 4 + 9 - 2 \cdot 2 \cdot 3 \cdot 0.5 \][/tex]
[tex]\[ c^2 = 13 - 6 \][/tex]
[tex]\[ c^2 = 7 \][/tex]
To find [tex]\( c \)[/tex], we take the square root of [tex]\( c^2 \)[/tex]:
[tex]\[ c = \sqrt{7} \][/tex]
Therefore, the length of the third side is:
[tex]\[ \boxed{\sqrt{7}} \][/tex]
