To determine the length of the third side of a triangle when you have two sides and the included angle, we use the Law of Cosines. The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Given:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( C = 60^\circ \)[/tex]
We need to calculate side [tex]\( c \)[/tex].
Firstly, we convert the angle [tex]\( 60^\circ \)[/tex] to radians because the cosine function commonly uses radian input:
[tex]\[ 60^\circ = \frac{\pi}{3} \text{ radians} \][/tex]
Next, we apply these values to the Law of Cosines formula. Plugging them in:
[tex]\[ c^2 = 3^2 + 4^2 - 2 \cdot 3 \cdot 4 \cdot \cos\left(\frac{\pi}{3}\right) \][/tex]
We know that:
[tex]\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
So substituting this in we get:
[tex]\[ c^2 = 9 + 16 - 2 \cdot 3 \cdot 4 \cdot \frac{1}{2} \][/tex]
[tex]\[ c^2 = 25 - 12 \][/tex]
[tex]\[ c^2 = 13 \][/tex]
Finally, taking the square root of both sides to solve for [tex]\( c \)[/tex]:
[tex]\[ c = \sqrt{13} \][/tex]
Therefore, the length of the third side is:
[tex]\[ \boxed{\sqrt{13}} \][/tex]
So, the correct answer is:
B. [tex]\(\sqrt{13}\)[/tex]
